Etale Homotopy of Simplicial Schemes. (AM-104), Volume 104 9781400881499

Table of contents :
INTRODUCTION
1. ETALE SITE OF A SIMPLICIAL SCHEME
2. SHEAVES AND COHOMOLOGY
3. COHOMOLOGY VIA HYPERCOVERINGS
4. ETALE TOPOLOGICAL TYPE
5. HOMOTOPY INVARIANTS
6. WEAK EQUIVALENCES, COMPLETIONS, AND HOMOTOPY LIMITS
7. FINITENESS AND HOMOLOGY
8. COMPARISON OF HOMOTOPY TYPES
9. APPLICATIONS TO TOPOLOGY
10. COMPARISON OF GEOMETRIC AND HOMOTOPY THEORETIC FIBRES
11. APPLICATIONS TO GEOMETRY
12. APPLICATIONS TO FINITE CHEVALLEY GROUPS
13. FUNCTION COMPLEXES
14. RELATIVE COHOMOLOGY
15. TUBULAR NEIGHBORHOODS
16. GENERALIZED COHOMOLOGY
17. POINCARÉ DUALITY AND LOCALLY COMPACT HOMOLOGY
REFERENCES
INDEX

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Etale Homotopy of Simplicial Schemes

INTRODUCTION In the early part of this century, alg eb raic geometry and alg eb raic topology were not se p a ra te d isc ip lin e s . Indeed many manifolds admit the structure of a (com plex) alg eb raic v ariety .

S. L e fs c h e tz and others intro­

duced algeb ro-geom etric techniques to study to p o lo g ical properties of such v a rie tie s. Subsequently, alg eb raic geometry widened its sco p e to include v a rie tie s of p ositiv e c h a ra c te r is tic , rings arisin g in alg eb raic number theory, and more general sch em es. An alg eb rizatio n of alg eb raic geometry has been ach iev ed , thanks in large part to A. Weil and 0 . Z a risk i; and powerful techniques of hom ological algebra have been employed, e s p e cia lly th ose introduced by J . - P . Serre and A. G rothendieck. On the other hand, alg eb raic topology has developed in d irectio n s not obviously relevant to alg eb raic geometry. Inspired by A. W eil's celeb rated co n je ctu re s, M. Artin and A. Grothen­ d ieck developed e ta le cohomology theory, a theory su fficien tly well behaved that in some w ays it plays the role for p o sitiv e c h a ra c te ris tic v a rie tie s of singular cohomology theory in alg eb ra ic topology.

P . D elig n e’s

proof of the Weil C onjectu res [22] is a dram atic application of e ta le cohom ology.

M. Artin and B . Mazur refined e ta le cohomology theory in

their book entitled E ta le Homotopy [8]:

they introduced a “ pro-homotopy

ty p e " a s s o c ia te d to a schem e whose cohomology is the e ta le cohomology of the sch em e. T hu s, e ta le homotopy theory reinforces the relationship betw een algeb raic geometry and alg eb raic topology. The purpose of th is book is to provide a coh eren t acco u n t of the current s ta te of e ta le homotopy theory. We d escrib e in C hapters 9 , 1 1 , and 1 2 , various ap p licatio n s of this theory to alg eb raic topology, cohomology of groups, and alg eb raic geom etry. T h e se ap p licatio n s have required 3

4

E T A L E HOMOTOPY O F SIM PLIC IA L SCHEMES

repeated refinement and g en eralizatio n of the theory developed by M. Artin and B . Mazur. F o r this reaso n , we con sid er sim p licial sch em es throughout, n e ce ssa ry for alm ost a ll e x istin g ap p lica tio n s, and introduce the e ta le top o lo g ical type which refines the e ta le homotopy type. With an eye to future a p p lica tio n s, we introduce various alg eb raic in­ varian ts of the e ta le top o lo g ical type not usually considered by alg eb raic geom eters:

function co m p lexes, re la tiv e homology and cohom ology, and

generalized cohom ology. It is our hope that this p resentation of e ta le homotopy theory w ill enable ap p licatio n s by to p o lo g ists and geom eters who have not immersed th em selves in the d e lic a te te c h n ic a litie s of the su b je ct. C onsequently, we have attem pted to minimize p re-requ isites to b a sic hom ological algebra [1 7 ], elem entary theory of sim p licial s e ts [57], and an acq u ain tan ce with alg eb raic geometry [50]. We have taken th is opportunity to re-work much of the foundational m aterial of e ta le homotopy developed by M. A rtin-B . Mazur and the author:

the knowledgeable reader

will reco g n ize many improvements and g en eralizatio n s of resu lts in the literatu re. We now proceed to briefly sk etch the con ten ts and organization of the various ch a p te rs, e a ch of which a lso has its own introduction. Beginning with the definition of a sim p licial sch em e, Chapter 1 d e scrib e s exam ples most common in ap p licatio n s and briefly d is c u s s e s a technique for ad apt­ ing co n stru ctio n s on sch em es to apply to sim p licial sch em es. Chapter 2 defines (e ta le ) sh eaf cohomology in terms of derived functors and re la te s the sh eaf cohomology of a sim p licial schem e to that of its con stituen t sch em es. A fter d iscu ssin g the somewhat more fam iliar C ech co n stru ctio n , we d escrib e in Chapter 3 the analogous con stru ctio n of J .- L . Verdier which en ables one to compute cohomology in terms of h ypercoverings. Chapter 4 p resen ts the e ta le to p o lo g ical type of a sim p licial schem e based on the in verse system of rigid hypercoverings (in itially inspired by work of S. Lubkin [54]). T his e ta le to p o lo g ical type is a refinement of the Artin-Mazur e ta le homotopy type and is given by a somewhat more natural co n stru ctio n than previous refinem ents. In Chapter 5 , we verify that the

5

INTRODUCTION

s e t of con n ected com ponents, the fundamental group, and the cohomology with lo cal co e fficie n ts of the e ta le to p o lo g ical type of a sim p licial schem e are given by the s e t of co n nected com ponents, the Grothendieck fundamen­ tal group, and the sh eaf cohomology of the sim p licial schem e. B e c a u s e the e ta le top o lo g ical type is not a sin g le sim p licial s e t but an in verse sy stem , m achinery must be employed to an aly ze its homotopy in­ varian ts. Chapter 6 com pares co n stru ctio n s of M. A rtin-B . Mazur, D. Sullivan, and A. K. B ousfield -D . M. Kan, e a ch of which has been em­ ployed in the literatu re for various a p p lica tio n s. To a s s i s t in the id en tifi­ catio n of various homotopy invariants using th e se co n stru ctio n s, we prove se v e ra l fin iten ess properties in Chapter 7. Chapter 8 then re la te s the e ta le top ological type of a sim p licial com plex variety to the geom etric realizatio n of its a s s o c ia te d sim p licial (a n a ly tic ) s p a c e and provides a com parison of top o lo gical types in c h a ra c te r is tic

p ^ 0 and

0.

An e sp e cia lly important

exam ple is that of the cla ssify in g s p a c e of a com plex red u ctive L ie group. Having travelled so far, the reader is rewarded with various a p p lica ­ tio n s. In terest in e ta le homotopy theory w as aroused by D. Q uillen’s use of the theory in his sk e tch of a proof of the (com plex) Adams co n jectu re [61].

We present a m odification of D. S u llivan ’s proof of the Adams co n ­

jectu re [69] as well a s our proof of its infinite loop sp a c e refinement [37]. Chapter 9 a lso d e scrib e s the au th or’s use of homomorphisms of alg eb raic groups in p ositive c h a ra c te r is tic to provide in terestin g maps of c la s s ify in g s p a c e s and homogeneous s p a c e s of com p act L ie groups [33], [36]. Chapter 11 p resen ts four geom etric ap p licatio n s of e ta le homotopy theory originally due to D eligne-Sullivan [25], D. Cox [20], and the author [3 0 ], [31].

The

la s t two ap p licatio n s of Chapter 11 and all th ose of Chapter 12 require a com parison of homotopy types of the algeb ro-geom etric and homotopy fibres a s d iscu sse d in Chapter 1 0 .

In Chapter 1 2 , we d escrib e further

ap p lication s by the author of e ta le homotopy to K -theories of finite field s and the cohomology of finite C hevalley groups [32], [34], [35], [41]. The d iscu ssio n of function com p lexes in Chapter 13 is cen tral to recen t ap p licatio n s of e ta le homotopy to alg eb raic K-theory by the author

6

E T A L E HOMOTOPY O F SIM PLIC IA L SCHEMES

and W. Dwyer [27], [39].

B e c a u s e this is the first published acco u n t of

function com p lexes, Chapter 13 exam ines their behavior in some d e ta il. In Chapter 1 4 , we d is c u s s re lativ e cohomology in order to incorporate lo c a l cohomology and cohomology with supports in the framework of e ta le homotopy theory. Chapter 15 provides a refinement of D. C o x ’s co n s tru c ­ tion of tubular neighborhoods [18] and u tiliz e s th e se tubular neighborhoods in a d iscu ssio n of e x c is io n and M ayer-V ietoris. Chapter 16 provides a first introduction of generalized cohomology into the study of alg eb raic geom etry: an early v ersion of this chapter was the b a sis for the au th or’s com parison of alg eb raic and to p o lo g ical K-theory of v a rie tie s [38].

F in a lly , Chapter

17 p resen ts a sk etch of P oin care' duality for e ta le cohomology using the machinery developed in the preceding ch a p te rs. During the lengthy evolution of th is book, further developm ents and ap p lication s of e ta le homotopy theory have a rise n . We refer the reader to [73] for a d iscu ssio n of sh e a v e s of s p e ctra and e ta le cohom ological d escen t u tilized in a further ap plication to a lg eb raic K-theory. We are e s p e c ia lly indebted to D. C ox, who shared with us pre-prints of his work on the homotopy type of sim p licial sch em es [19] and on tubular neighborhoods [18].

We a lso thank A. K. B o u sfield , W. Dwyer, and

R . Thom ason for many valuable co n v e rsa tio n s. F in a lly , we thank Oxford U n iversity, the U niversity of Cambridge, and the Institute for Advanced Study for their warm h osp itality during the writing of this book, and g rate­ fully acknow ledge support from the N ational S cie n ce Foundation and the S cien tific R e se a rch C ouncil.

1.

E T A L E SITE OF A SIMPLICIAL SCHEME

After statin g the definition of a sim p licial sch em e, we provide s e v e ra l important exam ples in E xam p les 1 .1 , 1 .2 , and 1 .3 .

The first two exam ples

a rise frequently in ap p lica tio n s, w hereas the third is cen tral in the co n ­ struction of the e ta le homotopy type. e ta le site of a sim p licial sch em e.

In D efinition 1 .4 , we introduce the

As we exp lain , the e ta le s ite is a

gen eralization due to G rothendieck of the categ o ry of open su b se ts of a top olo gical s p a c e . We conclude this chapter with a co n stru ctio n which en ables one to extend ce rta in arguments applied to a given dimension of a sim p licial sch em e to the entire sim p licial sch em e. We re ca ll th at A , the ca teg o ry of standard s im p lic e s , c o n s is ts of o b jects

A[n] for n > 0 and maps

function from | 0 ,l,---,n ! to

A[n] -> A[m] for e a ch nondecreasing

{ 0 , 1 , •••,m|. Any such map a : A [ n ] A [ m ]

(other than the id entity) can be written as a com posite of “ d e g e n e ra cy ” maps

A[k—1 ] with j < k -1

o su rje ctiv e and “ f a c e ” maps

defined by

A [£+l] with i < £+1 defined as

that in jectiv e map su ch that i e {0,•••,£+! \ is not in the image of d- . A sim p licia l o b ject of a categ o ry opposite categ o ry to

A;

C is a functor A 0 -» C ,

where A 0 is the

a map of sim p licial o b jects is a natural tra n s­

formation of functors. A sim p licia l s c h e m e is a functor from A 0 to the categ o ry of sch em es ( = lo ca l ringed s p a c e s which admit a covering by open su b sp a ce s isom or­ phic to the sp ectra of rings; c f ., [50], II.2).

Follow ing conventional n o ta­

tion, we denote such a sim p licial sch em e by X . , and we denote X .(A [n ]) by X n . E X A M P L E 1 .1 . X®T.

Let

X

be a schem e and

is the sim p licial schem e with

T.

a sim p licial s e t.

Then

( X ® T .) n equal to the disjoint union 7

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

8

indexed by T n and with a : ( X ® T .) m -> (X ® T .)n for

of co p ies of X

a : A[n] -» A[m] in A given by sending the summand of X

indexed by

t e T m via the identity to the summand of X

indexed by a (t) e T n . More

gen erally, if X .

T.

X .® T .

is a sim p licial schem e and

a sim p licial s e t, then

is sim ilarly defined with (X . ® T .)n equal to the d isjoin t union of

co p ies of X n indexed by T n . Let

A[k] a ls o denote the sim p licial s e t defined by A [k]n =

H om ^(A[n], A [k]) . In p articu lar, is equal to X

in e a ch dimension (we sh a ll usually denote

simply by X ). X ® A[k] -> Y .

X ® A [0 ] is the sim p licial schem e which X ® A [0]

We e a sily verify that a map of sim p licia l sch em es

is eq uivalent to a map of sch em es

X

. A sp e c ia l c a s e

of Exam ple 1.1 which we sh all frequently employ is that in which T. equals

A [1 ] s o that

(X .® A [ l ] ) n is the d isjoin t union of n+1

X n . Two maps f,g : X . -> Y .

of sim p licial sch em es are said to be related

by the sim p licia l homotopy H : X .® A [l] -»Y . tions of H to X .® A [0 ] are If X

f and

if the two can o n ical re s tric ­

g.

is a schem e which appears a c y c lic with re sp e ct to a ce rta in

cohomology theory (e .g .,

X might be the spectrum of an a lg eb raically

clo sed field ), then X ® T . c ia l s e t

co p ies of

T.

appears to be an alg eb raic model of the sim pli­

for this cohomology theory.

T his is exploited in Chapter 11

to study finite C hevalley groups, as well a s by Z . Wojtkowiak in [72], and (im p licitly) by C. Soule in [67].

U sing the s p e c ia l c a s e

T = B Z /2 ,

D. Cox has studied real alg eb raic v a rie tie s (s e e C orollary 1 1 .3 ). ■ S be a sch em e.

A group s c h e m e over S , G ,

E X A M P L E 1 .2 .

Let

is a

map of sch em es

G -» S with the property that Homs ( ,G ) is a group

valued functor on the categ o ry of sch em es over S (i .e ., maps of sch em es Z -> S ).

In other words,

p : G xs G ^G

G is provided with maps

over S sa tisfy in g the usual axiom s.

e : S -> G and If G is a group

schem e over S , the c la s s ify in g s im p licia l s c h e m e BG is the sim p licial schem e given in dimension

n > 0 by (®G )n = Gx n , the n-fold fibre

product of G with itse lf over S , and with stru ctu re maps given in the

9

1. E T A L E S IT E OF A SIM PLIC IA L SCHEME

usual way using e

and

/z.

More gen erally, if X

schem e over S provided with left G -action G -action

(re sp e ctiv e ly ,

a: G xg X ^ X

Y ) is a

(re s p ., right

/3 : Y x g G -> G ), then B (Y ,G ,X ) is the sim p licial schem e given

in dimension n > 0 by B (Y ,G ,X )n = Y x s Gx n x s X and with stru ctu re maps given in the usual way using e , [jl , a and

/3

(c f ., [58], for an e x p licit d escrip tion of this “ d o u b le bar co n stru ctio n ” ). We let B (Y ,G ,* ) (re s p .,

B (* ,G ,X ) ) be the sim p licia l sch em es obtained

from B (Y ,G ,X ) by deleting the right-hand (re sp ., left-hand) facto r. S pecial c a s e s of Exam ple 1 .2 have occurred in most of the a p p lica ­ tions of e ta le homotopy theory. T h ese exam ples are so prevalent b e ca u se any com plex red u ctive L ie group G (C) adm its the structure of an a lg e ­ b raic group G^, over Spec C; BG^, then provides an alg eb raic model for the top ological c la ss ify in g sp a ce of G (C ). Moreover, if H is a subgroup sch em e of G over S , then H a c ts on G by m ultiplication so that B (G ,H ,*) and B (*,H ,G ) se rv e a s models for the “ homogeneous s p a c e ” G /H

(se e Chapter 9 ). ■

E X A M P L E 1 .3.

Let

For

x / n ^ denote the n-th truncation of X . : x / n ^ is the

n > 0,

let

restrictio n of X . F o r any

n > 0,

(or c o s k nX .

X.

be a sim p licial schem e over a given schem e

to the fu ll-su bcategory of A0 with o b jects we define the n-th c o s k eleto n of X .

S.

A[k], k < n .

over S, co sk ^ X .

if no confusion a ris e s by leaving S im plicit) by the follow ­

ing universal property:

F o r any sim p licial sch em e

of maps Y . -> co sk ^ X .

over S is in natural one-to-one correspon d ence

with the s e t of maps Y / n^ -> x / n ^ over S .

Y.

over S ,

the se t

If we view co sk ^ ( ) as a

functor from n-truncated sim p licial sch em es to sim p licial sch e m e s, then cosk ^ ( If X

) is right adjoint to the n-truncation functor ( )^n\ is a schem e over S, cosk ^ X

n erv e (u su ally w ritten Ng(X ) ) of X

( = co sk ^ (X ® A [0]) ) is the C e c h over S with

(co sk g X )n equal to

the (n-fl)-fold fibre product of X with its e lf over S . One v erifies that

10

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

co sk ^ X . k S p e c k ,

and a sh eaf F

of ab elian groups on E t(S p ec k) is

equivalent to the data of the group co lim F (S p ec K -►Spec k) = F (k ) provided K /k with a left actio n of the galois group lizer of any elem ent of F (k )

lim G a l(K /k ) such that the sta b iK /k

is a subgroup of finite index ([5 9 ], II. 1 .9 ).

In p articu lar, if k is a separably (a lg e b ra ica lly ) clo se d field, than a sh eaf F

on E t(S p ec k) is equivalent to its value F (S p e c k ) . Viewing

such S p eck

as “ p o in ts” for the e ta le topology, we define a g eo m etric

point of a sim p licial schem e X .

to be a map Spec k->Xm ,

some

m> 0

2. SH EA VES AND COHOMOLOGY

and some sep arab ly clo se d field

k.

15

We define the sta lk of a sh eaf F

on

E t(X .) at the geom etric point a : S p e ck -> X m to be a * F m(S p e c k ). We proceed to the definition of the (e ta le ) cohomology of an a b elia n s h e a f ( i .e ., a sh e a f of ab elian groups) F

on E t ( X .)

using derived

fu n ctors. F o r a more detailed d is cu ssio n , we refer the reader to [59]. L et X.

PROPOSITION 2 .2 .

be a s im p licia l s c h e m e , and let AbSh(X.)

den o te the ca tego ry of a b elia n s h e a v e s on E t ( X .) .

T h en A bSh(X .)

is an

a b elian ca tego ry with enough in je c tiv e s . M oreover, a s e q u e n c e of s h e a v e s in A bSh(X .) X m of X .

is e x a c t if and only if for ev ery g eo m etric point a : Spec k -*

the s e q u e n c e of sta lk s at a

P ro o f. A map A cokernel

is ex a ct.

in A bSh(X .) has kernel

B -> C ) if and only if the re strictio n

K^A

(re sp e ctiv e ly ,

Kn -> An (re s p .,

B n ^ C n)

is the kernel (re sp ., cokernel) of A n -» B n in AbSh(Xn) for e a ch

n > 0,

where AbSh(Xn) is the ab elian categ o ry of ab elian s h e a v e s on E t(X n) (cf. [59], II.2 .1 5 ). categ o ry .

T his readily im plies that A bSh(X.) is an ab elian

T o show

A bSh(X .) has enough in je c tiv e s , we use the functor

R n( ) : AbSh(Xn) -> A bSh(X .) defined by sending G e AbSh(Xn) to R n(G) with (R „(G ))rn = n m

a ^. G .

II

B ecau se

R IIn( ) is right adjoint to the

A[nL restrictio n functor (which is e x a c t),

R n(G) is in je ctiv e in A bSh(X.)

whenever G is in je ctiv e in AbSh(Xn) . T hu s, if F

is an arbitrary

abelian sh eaf on E t(X .) and if Fn -> I in jectiv e

In of AbSh(Xn) for ea ch

monomorphism of F

is a monomorphism of F n in an CO n > 0 , then F -» II R n(In) is a

into an in jectiv e of A b S h (X .).

n=0

B e c a u s e a seq uen ce

of sh eav es in A bSh(X .) is e x a c t if and only if its re strictio n to e a ch AbSh(Xn) is e x a c t, the la s t statem ent follows from the corresponding fact for AbSh(Xn) (cf. [5 9 ], I I .2 .1 5 ). ■ The following definition of the (e ta le ) cohomology of a sim p licial schem e (due to P . D eligne in [23], 5 .2 ) is the natural gen eralization of the

16

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

definition of cohomology of a schem e as derived functors of the global se ctio n functor. DEFINITION 2 .3 .

Let X.

be a sim p licial sch em e.

F o r any

i>0,

the

cohom ology group functor Hi (X. , ) : A bSh(X .) -> Ab is the i-th right derived functor of the functor sending an abelian sh e a f F on E t( X .)

to the ab elian group given a s the kernel of the map

d * - d * : F ( X 0) - ^ ( X ^ .

E q u iv alen tly ,

H i(X . , ) = E x t ^ b sh (x> )(Z , where to

)

Z is the co n stan t ab elian sh eaf on E t(X .) with e a ch sta lk equal

Z. ■ The definition of H *(X . ,F )

is global in the s e n se th at it is given in

terms of E t(X .) rather than the various

E t ( X n) .

N on eth eless, the follow ­

ing proposition en ables us to re la te H *(X . , F ) to the various H *(X n,F n), n > 0 . PROPOSITION 2 .4 . F e A b S h (X .).

L et X.

b e a sim p licia l s c h e m e , and let

T h en there e x is t s a first quadrant s p e ctra l s e q u e n c e E f ’t = ^ ( X g .F g ) => Hs + t(X . ,F )

natural with r e s p e c t to X . Proof.

Let

and F .

F -> I’ be an in jectiv e resolution in A b S h (X .).

We con sid er

the functor L n( ) : AbSh(Xn) -> A bSh(X .) defined by sending G e AbSh(Xn) to L n(G) defined by ( L n(G ))m =

© a*G , any n > 0 . aeA[m]n

B ecau se

L n( ) is an e x a c t left adjoint to the re strictio n functor ( ) n , we co n ­ clude that

( )

sends in je ctiv es to in je ctiv e s for e a ch

Fn -» (I*)n is an in jectiv e resolution in AbSh(Xn) .

n > 0.

T hus,

17

2. SH EAVES AND COHOMOLOGY

Let

Zx

denote the com plex of ab elian sh e a v e s in A bSh(X.) * whose m-th term is the ab elian sh eaf represented (in the sm all) by id : X m -> X m in E t ( X m) (and whose differen tial is the usual alternating sum of the maps induced by d- : X m -> X m_ 1 , i < m ). L m(Z|x ) ; when re stricte d to ' m

Zx

-> Z by defining Z Q(U) = *

Z.

We define an augmentation

© Z (U ) -> Z(U ) for any A [n] 0

in E t( X .) to be the identity on e ach summand. is e x a c t in A b S h (X .),

Z m equals

E t(X n) , Z m is the d irect sum indexed by

A[n]m of copies of the con stan t sh eaf map

T hus,

To verify that

U -» X n Zx

-> Z->0

*

it su ffice s to verify that the re strictio n s

(Z x )n Z -> 0 are e x a c t in AbSh(Xn) for n > 0 . This e x a c tn e s s follows * from the a c y c li c ity of A[n] and the identification of ( Z x ) n -> Z with Z® (A[n] -> A [ 0 ] ) . The sp e ctra l seq uen ce is obtained from the bi-com plex ^ omAbSh(X )( ^ x H °mA bsh(X

to tal cohomology of the bi-com plex is that of which is

H *(X . , F ) ;

w hereas the cohomology of

H° mA b S h ( X . ) ( V n = Ho mAb s h (x m)(Z >(Om) is

H *(X m, F m) . -

As an immediate corollary of P rop osition 2 .4 , we conclude that if F c A bSh(X .) is su ch that F n e AbSh(Xn) is in je ctiv e for e ach then F

is a c y c lic ( i .e .,

co ch ain com plex

n > 0,

H \ X . ,F ) = 0 for i > 0 ) if and only if the

F ( X .) = In h> F ( X n)i is a c y c li c .

We sh all often u tilize b i-sim p licial sch em es to study sim p licial sch em es.

As the reader can s e e from the following brief d iscu ssio n , the

e tale topology and e ta le cohomology of b i-sim p licial sch em es are defined analogously to that of sim p licial sch e m e s.

Moreover, P rop osition 2 .5

permits us to rep la ce a b i-sim p licial schem e c ia l sc h e m e A X .,

X ..

by its diagonal sim pli-

(where (A X ..) n = X n n) .

A b i-sim p licia l s c h e m e is a functor from A0 x A0 to the categ o ry of sch em es.

Follow in g convention, we denote su ch a b i-sim p licial schem e

by X .. , and we denote X ..(A [m ], A[n]) by X m n . We let

E t ( X ..)

denote

18

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

the category whose ob jects a r e retale maps

U

X g ^ for some s , t > 0

and whose maps are commutative squares U

►V

X s , t ---------------------" x k,£ whose bottom arrow is a specified structure map of X .. . We give

E t(X ..)

the etale topology by defining a covering of U -> X c f to be a family of over X g ^ whose images cover

etale maps

a lso denote the resultant e ta le site on the category

U.

We let E t ( X . . )

E t(X ..) .

We define A bSh(X..) to be the abelian category of abelian s h e a v e s on the s it e

E t(X ..),

where the sh eaf axiom is that of Definition 2 . 1.

(e ta le ) cohom ology groups of a b i-sim p licia l s c h e m e X . . sheaf F e AbSh(X..)

The

with values in a

are defined to be

H \ X . , F ) = E < bsh(x> )( Z , F ) where

Z is the con stan t abelian sh eaf on E t ( X . ) with fibres

PROPOSITION 2 . 5 .

L e t X ..

be a b i-sim p licia l s c h e m e .

Z.

T h en th ere is a

natural isom orphism of 8 -functors H *(X .. , ) ^ on A b S h (X ..), and (

w here A X .,

H *(A X .. ,(

)A)

is the diagonal (s im p licia l sch e m e ) of X . . ,

: AbS h (X ..) -» AbSh(AX..)

is the restrictio n functor.

P roof. It su ffices to (a ) exhibit a natural isomorphism H ° ( X .. , ) H ° ( A X .. ,(

)A) ,

(b) prove H ^ A X .. ,IA) = 0 for i > 0

in je ctive , and (c) verify that H *(A X .. ,(

any

I e AbSh(X ..)

)A) is in fact a 8 -fu n cto r (i .e .,

there is naturally a s s o c ia t e d a long e x a c t seq uen ce

19

2. SH EA VES AND COHOMOLOGY

••• -> H '(A X .. , F A) -> Hi(A X .. ,G A) -> H ^ A X .. ,HA) -> Hi+1(A X .. , F A) - ••• to each short e x a c t seq u en ce V. 4 .4 ).

0^F^G->H->0

in A b S h (X ..)) (c f ., [17],

P a rt (a ) is readily ach ieved by identifying both H ° ( X ..,

) and

H °(A X .. ,( ) A) with the group K e r ( d £ ° d * - d * ° d * : F ( X 0 0) - F ( X 1(1) ) . Arguing as in the proof of Prop osition 2 .4 , we verify that is in jectiv e whenever I e A b S h (X ..) is in je ctiv e . su ffice s to prove that

I^ (A X ..) == {n k I(X n

I e A b S h (X ..) is in je ctiv e .

B ecau se

)}

Ig t e AbSh(Xg t)

Thus, to prove (b), it

is a c y c lic whenever

T o t(I (X ..)) and A l(X ..) = IA(A X ..)

have the sam e cohomology ([2 6 ], 2 .9 ), this follows from the e x a c tn e s s of T o t(Z x

) whose proof is analogous to that of the e x a c tn e s s of

given in P rop osition 2 .4 . ( )^

is e x a c t b e ca u se

only if 0

F g t ^ Gg t

Zx

F in a lly , to prove (c ) , it su ffice s to observe that 0^F-*G-*H-*0

is e x a c t in A b S h (X ..) if and

Hg t ^ 0 is e x a c t in AbSh(Xg t) for all

s ,t > 0 . ■ We conclude this ch apter with the following b i-sim p licial analogue of P roposition 2 .4 . PROPOSITION 2 .6 .

L e t X ..

a b elia n s h e a f on E t ( X . . ) .

b e a b i-sim p licia l s c h e m e and let F

b e an

T h en there e x is t s a first quadrant s p e c tra l

sequ en ce E f ' 1 = H ^ X ^ ,F Sm) = > Hs + t(X .. ,F ) natural with r e s p e c t to X ..

and F .

P ro o f. Em ploying L s * ( ) : AbSh(Xg )-> A b S h (X ..) as

in the proof of

P rop osition 2 .4 , we conclude that Ig e AbSh(Xg ) is in jectiv e whenever I 6 AbSh(X..)

is in je ctiv e .

We define the com plex Z x of sh e a v e s on

E t ( X ..) with n-th term equal to L n* (Z ) e A b S h (X ..) , so th at Z Y is a resolution in A b S h (X ..). a s s o c ia te d to the bicom plex

-^Z

The a s se rte d s p e ctra l seq u en ce is that H °mAbsh(X

an in jectiv e resolution in A b S h (X ..). a

)( ^ x

where F -> I* is

3.

COHOMOLOGY VIA HYPERCOVERINGS

The main resu lt of this ch apter is Theorem 3 .8 which a s s e rts that sh eaf cohomology of a sim p licial schem e ca n be computed in a som ew hat com binatorial way using h ypercoverin gs. B e c a u s e the definition of a hyper­ covering and the required properties of the categ o ry of hypercoverings are somewhat formidable at first encounter, we begin this ch apter with a d iscu ssio n of the sim pler co n te x t of C ech nerves and C ech cohom ology.

As

seen in C orollary 3 .9 , C ech cohomology is naturally isom orphic to sh eaf cohomology in many c a s e s of in terest (this is a very s p e c ia l property of the e ta le s ite ).

By considering h ypercoverin gs, we provide com binatorial

arguments ap p licab le to any s ite only notationally more com plicated than th ose of C ech theory (s e e , for exam ple, Lemma 8 .3 ). be a sim p licial sch em e. An (e ta le ) co v erin g U. -> X . of X . \/ is an e tale su rje ctiv e map. The C e c h n e rv e of U. -> X . is the b isim p licial Let

schem e

X.

Nx (U .) defined by

Nx . ( u " ) s ,t = (N x s (Us ))t the (t+ l)-fold fibre product of U s with its e lf over X g (s e e Exam ple 1 .3 ). F o r any a b elia n p re s h e a f P define s ,t

( i .e ., functor E t ( X .) °

(a b .g rps) ), we

P (N X (U .)) to be the b i-coch ain com plex given in bi-codim ension

by P (N X (U ..)S f.) with d ifferen tials obtained in the usual way a s an

altern atin g sum of the maps

d * . We re c a ll that the cohomology of su ch a

bi-com plex is the cohomology of the a s s o c ia te d to tal com plex b e a sim p licia l s c h e m e and P an a b elia n \/ s h e a f on X . . For any i > 0 , d e fin e the C e c h cohom ology of X . with

PROPOSITION 3 .1 .

v a lu es in P

L et X.

in d e g re e i by

20

3. COHOMOLOGY VIA H YPERC O V ER IN G S

21

= colim H^PCNy (U .)))

u.^x.

A*

w here the colim it is in d ex e d by c o v erin g s U. -> X . of X . So d efin e d , v* H (X , ) is a 8 -functor on the ca teg o ry of a b elia n p re s h e a v e s on X . . P ro o f. L e t

U. -» X .

and V. -> X .

be co v erin g s of X .

defined by (U. x x V .)n = ( ^ n x x more, if f,g : U. -> V.

Then

*s a ^so a covering of X .

colim

F u rth er­

are two maps over X . , then

Nx . ( f )* = Nx . ( g ) * : H *(P (N X (V .))) - H *(P (N X>(U .))) T hus,

U. x x V. ->X.

(s e e below).

can be re-indexed by the opposite categ o ry to the left

U.->X.

d ir e c te d ca tego ry ( i .e ., category with the properties that there is at most one map between any two o b je cts and th at for any two o b jects there e x is ts a third mapping to both) of coverin gs of X . maps, and h ence is e x a c t. f,g : U. -> V.

and eq uivalen ce c l a s s e s of

The fa c t that Nx (f )* = Nx (g )*

is proved using the fa ct that two maps

for

fn,g n ' Un -> Vn over

X n for n > 0 have the property that

c o s k 0 n(fn), c o s k 0 n(g n) : c o s k 0 n(Un) -> c o s k Qn(Vn) X

are related by a unique sim p licial homotopy (sin ce a map c o s k Q (Un) ® A [1 ]

xn

-> c o s k Q (Vfl)

is equivalent to its re strictio n to dimension

0 ).

T hu s,

Nx ( f ) and Nx (g) are related by a b i-sim p licia l homotopy NX . (U .)® (A [0 ]x A [ l]) -» Nx _ (V .), where Q*X t =

s o th at Nx _ (f )* = Nx > (g ) * .

By definition, an e x a c t seq u en ce of ab elian p re-sh eaves P3

0 on E t(X .)

P3(U) -» 0

is a seq u en ce with the property that

is e x a c t for every

U -> X n in E t ( X .) .

0 - P l - P2 -

0 -> P j ^ ) -> P2 (U)

T hu s, such a short e x a c t

seq uen ce induces a short e x a c t seq u en ce of bi-com plexes

0 -> P j(N x (U .)) ->

P2(N x (U .)) ^ P 3(N x (U .)) -> 0 for any U. -> X ; this short e x a c t seq uen ce yields a long e x a c t seq uen ce

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

22

ftJ.))) — • By e x a c tn e s s of colim as d iscu sse d ab ove, we conclude the long e x a c t U.^X. seq u en ce ••• -* H ^ X . ,P j ) In other w ords,

H*(X. ,P 2> - H ^X . ,P 3) - Hi+1(X . .P ^ -» ••• .

V* H (X . , ) is a 5-fun ctor. ■

As we will s e e in the next proposition, there is a s p e ctra l seq u en ce analogous to th at of Prop osition 2 .4 which re la te s C ech cohomology of X . with that of each of the X PROPOSITION 3 .2 . p re s h e a f on X . .

L et X.

, n > 0. b e a sim p licia l s c h e m e and P

an a b elia n

T h en there e x is t s a first quadrant s p ec tra l s e q u e n c e

E f ' 1 = i W x ^ P g ) = > Hs + t(X . ,P ) w here Pg

is the restrictio n of P

P ro o f. F o r any covering to the bicom plex

to E t(X g ) .

U. -> X . , there is a sp e ctra l seq u en ce a s s o cia te d

P (N X (U .)) of the form

E ^ O J . ) = Ht(Pg(N x (Us ))) = > Hs + t(P (N x / U . » ) •

B e c a u s e two maps U. 3 V.

over X .

induce maps

P (N X (V .)) 3 P (N X (U .))

which are related by a filtration -p reservin g homotopy, we conclude that two su ch maps induce the sam e map

{E ^ C V O U

-

.

T h erefore, we may take the colim it of the s p e ctra l seq u en ce indexed by the left directed categ o ry of co v erin g s

U. -> X .

maps to obtain the sp e c tra l seq u en ce

and eq uivalen ce c l a s s e s of

23

3. COHOMOLOGY VIA H YPERC O V ER IN G S

E ® '1 = colim H^PgCNx (U s ))) = > Hs + t(X . ,P ) .

To conclude the proof of P ro p o sitio n 3 .2 , it su ffice s to verify for any s ,t > 0 that the natural map

colim H ^ C N x (U s ))) - H ^ X g .P g ) = colim H ^ P ^ x

(where the first colim it is indexed by coverin gs by coverin g s

W -> X g ) is an isomorphism.

U. = r * '( W s ) -> X .

and the second

F o r th is, it su ffice s to observe

that if W -> X g is e ta le and s u rje c tiv e , then W ^ X S where

U. -> X .

(W )»

Ug -> X g fa cto rs through

(c f ., Prop. 1 .5 ). ■

The failure of C ech cohomology to equal sh eaf cohomology a ris e s from the fa ct that a family of coverings U ’s

U -» X

becom e “ arbitrarily fin e ” while the

can have the property that the Nx (U )^ ’s

do not becom e arbi­

trarily fine for some k > 1 . F o r exam ple, the fa ct that the

U ’s

a c y c lic need not imply that the U x x U ’s

T his problem

becom e a c y c lic .

become

is circum vented by introducing h ypercoverin gs, the following g en eraliza\/ tion of C ech n erves. DEFINITION 3 .3 .

Let

co v erin g

is a b i-sim p licial schem e over

U .. -» X .

X.

An (e ta le ) h y p er­

be a sim p licial sch em e.

X.

with the property

that

UcO* -> X b is a hypercovering of X b for ea ch s > 0 ( i .e ., U Oj fL -» X X ( c o s k ^ U g )t is e ta le su rje ctiv e for all t > 0 , where c o s k j Ug = X g ). The homotopy categ o ry of hypercoverings of X . , denoted

categ o ry whose o b je cts are hypercoverings

U .. -> X.

eq u ivalen ce c la s s e s of maps of hypercoverings of X .

H R (X .),

is the

and w hose maps are ( i .e ., of b i-sim p licial

sch em es over X . ) where the eq uivalen ce relation is generated by pairs of maps U .. =» V ..

related by a sim p licial homotopy

U .. ® (A [0] x A [1 ]) -> V ..

over X . . ■ We re ca ll that the homotopy categ o ry of C ech nerves of coverings U. -» X .

is a left d irected cate g o ry .

A g en eralization of left directed

24

E T A L E HOMOTOPY OF SIM P LIC IA L SCHEMES

categ ory is that of a left filterin g ca teg o ry . A categ o ry

C is left filtering

if (i) for every pair of o b je cts

c " in C together

c , c ' in C ,

there e x is ts

with maps c «- c"-> c ' ; and (ii) for every pair of maps c ' 3 c e x is ts -> c

c " ^ > c ' in

C such that the co m p osites

in C , there

are equal ( c "

c " - > c '= > c

is called a left e q u a liz e r of c ' 3 c ). In the proof of Proposition 3 .2 we used the fa c t that every covering

W -» X g is dominated by Ug -> X g where

U. = r^ * (W ).

We g en eralize this

relation sh ip betw een the left d irected ca te g o rie s of coverings of X . coverin gs of X g a s follow s.

and

A functor F : C -> D is said to be left final

provided th at (i) for every o b ject d of D , there e x is ts an o b ject c C and a map F ( c ) every pair of maps

d in D ; and (ii) for every ob ject c F (c )3 d

in D ,

there e x is ts

that F (y ) is a left eq u alizer of the given maps

in

in C and

y:c'->c

in C such

F (c ) 3 d .

The u sefu ln ess of th e se g en eralizatio n s is that if C is left filtering, then the colim it indexed by the opposite categ o ry of C ,

colim , is an

c° e x a c t functor on the categ o ry of ab elian group valued functors on C ° ; m oreover, if

C and D are left filterin g,

F : C -> D is left final, and

P : D ° -> Ab is any functor, then the natural map colim P ° F an isomorphism (c f ., [8], A .1 .8 ).

c 0

colim P

is

^0

P rop osition 3 .4 re v e a ls the relev an ce of this d iscu ssio n to the categ o ry H R (X .) of Definition 3 .3 . PROPOSITION 3 .4 .

L et X.

be a sim p licia l s c h e m e .

left filterin g and the restrictio n map H R (X .) -> H R (X n)

T h en H R (X .)

is

is left fin a l for

ea ch n > 0 . P roof.

If U .. -> X .

and V .. -» X .

(defined by (U .. x x V ..) g t = U .. -» X .

are h ypercoverin gs, then t x X ^s t

and V .. -> X . . If U .. 3 V ..

U ..x -^ V ..-> X .

a hypercovering mapping to

are two maps of hypercoverings of

X . , we use the con stru ctio n of ([7 ], V .7 .3 .7 ) to co n stru ct the left eq u alizer. Namely, for ea ch Ws

s > 0,

there is a naturally defined sim p licial schem e

= H om (X s ® A [l],V g ) over X g with the property that

3. COHOMOLOGY VIA H YPERC O V ER IN G S

25

H om (Z .,H om (X s ® A [l],V SB)) - Hom (Z.® A [1],V S T.

is s p e c ia l over Y

if Z n maps su r-

je ctiv e ly onto the fibre product of ( c o s k ^ ^ Z . ^ -» ( c o s k ^ j T . ) 0 ), a s seen for exam ple in [29], 1 .3 .

product IK. of W.. -» V . . x x V .. U ..

is a left eq ualizer of U .. 3 V .. .

T o prove that H R (X .) -» H R (X n) is left final, we use P rop osition 1 .5 to conclude that

C ) sending

W. to

r^ * (W .) (defined by

(r n - ( W-))s ,t = ( r n '( Wt)> s) is right adjoint to the re strictio n functor from b isim p licial sch em es over X . if W.

to sim p licia l sch em es over X

is a hypercovering of X n ,

a hypercovering of X .

(b e ca u se

of pull-backs of W. -> X n ).

. M oreover,

then r ^ ‘(W.)is readily ch eck ed to be (r^ * (W .))

is the

fiber product over X g

T hu s, the left finality of H R (X .) -> H R (X n)

follow s from the following lemma (w hose proof we le a v e a s an e x e rc is e ). ■ LEMMA 3 .5 .

L e t F : C -» D b e a functor inducing a functor HoF : HoC ->

HoD w here maps in HoC a nd HoD a re e q u iv a le n c e c la s s e s of maps of C and D r e s p e c t iv e ly .

If HoC

adm its a right adjoint, then HoF

a nd HoD a re left

filterin g and if F

is left fin a l, m

T he next proposition in d icates that a hypercovering of a schem e should be viewed as a resolution of that sch em e. PROPOSITION 3 .6 . L et

Zjj

U. -> X

L et X

b e a s c h e m e and U. -> X

a hy p er co v erin g .

b e the co m p lex of a b elia n s h e a v e s in AbSh(X) d eterm in ed by * w hose m-th term is the a b elia n s h e a f re p r e s e n te d by Um -» X in

26

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

T h en the natural augm entation map Z u

E t(X ).

exp resses

X^

a s a reso lu tio n in AbSh(X)

in d u ced by U Q -» X

-> Z * of Z .

* P roof. T o prove that show that x * ( Z u

X^

-» Z -» 0 is e x a c t in A b S h (X ),

it su ffice s to

*

-»Z -»0) in AbSh(Spec k) is e x a c t for every geom etric *

point x :S p e c k - > X x :S p e c k -> X

(by P rop osition 2 .2 ).

and let W. = U. x x S p eck -> S p eck

U. -> X by x .

Then x * ( Z u

*

^ Z -> 0 ) eq uals

AbSh(Spec k ) . B e c a u s e E t(S p ec k) has o b ject, the global s e ctio n functor Y e t,

F ix a geom etric point denote the pull-back of

Zw *

prove that

AbSh(Spec k)->Ab

W. (ignoring the sch em e stru ctu re).

C ^ W .) -> Z -» 0 is e x a c t.

0 in

id : Spec k -»Spec k a s an in itial

Z w (Spec k) is the free ab elian chain com plex *

sim p licial s e t

Z

is an eq u iv alen ce. C ^ W .) on the T hus, it su ffice s to

T his follow s from the ob servation

that W. is the co n tra ctib le Kan com plex b e ca u s e it s a tis fie s the hyper­ covering condition that

^ (c o s k ^ W .)^

be su rje ctiv e (a s a map of s e t s )

for a ll t > 0 . ■ Using P rop osition 3 .6 , we prove that a sim p licial schem e and its h ypercoverings have the sam e cohom ology. PROPOSITION 3 .7 . hyper co v erin g. AbSh(X .)

L et X.

be a s im p licia l s c h e m e and U .. -> X .

T h en A U .. -> X .

be a

in d u ces an isom orphism of 8-fu n cto rs on

* _ H (X . , )

* H (U .. ,

) .

P roof. If F e A bSh(X .) , then H °(X . ,F ) = K e r (F (X Q) 5 F ^ ) ) , w hereas H °(U .. ,F ) = K e r(F (U 0>0) 5 F ( U l f l )) = K e r (d * -d * : F ( U 0 0) - F ( U l f l ) ) . B ecau se

U n -> X n is a hypercovering for e a ch

n > 0,

the sh eaf axiom

im plies th at K er(F (U n>0) 5 F (U n>1)) = K e r(F (U n>0) 5 F (U n>0x X n Un>0)) equals

F ( X n) .

C onsequently,

H °(U .. ,F ) = K e r(F (U 0>0) - F ( U 1 (1 )) - X K e r (F (X 0) 5 F ( X j ) ) = H °(X . , F ) .

3. COHOMOLOGY VIA H YPERC O V ER IN G S

27

Since the restrictio n functor A bSh(X .) -> A bSh(U ..) is e x a c t, is a 5-functor on A b S h (X .).

H *(U .. , )

T hus, it su ffice s to prove that H *(U .. ,1) =

H *(X . ,1) for I in je ctiv e in A b S h (X .).

B ecause

Us

is e ta le , the

restrictio n functor has an e x a c t left adjoint y : Ab(Us

-> A b (X g ) defined

by sending a sh eaf G on Ug t to the sh eaf a s s o c ia te d to the presheaf (V -»Xg ) k ©G(V ^ U g t ) , where the sum is indexed by the s e t of maps V -> Ug j. over X g . C onsequently, the re strictio n to Ug t of an in je ctiv e on X s

is in je ctiv e .

T herefore,

P rop osition s 2 .4 and 2 .5 .

H * (U .. ,1) = H *(A U .. ,1) - H *(I(A U ..)) by

To compute

H *(I(A U ..)) = H * (I(U ..)),

we

employ the sp e ctra l seq u en ce E f ’4 = H ^ C U ^ ) ) = > Hs + t(I(U ..)) . B ecau se

Ig is in jectiv e in AbSh(Xg ) and b e ca u s e

H om ^g^^

sjc>IS)»

Is (Ug ) =

Prop osition 3 .6 im plies that H*(IS(U .)) = 0 for

t > 0 and H °(Ig(U g )) = Is (X g ) for e ach H *(I(U ..)) = H *(I(X .)) which equals

s > 0.

C onsequently,

H *(X . ,1) by the remark following

P rop osition 2 .4 . ■ We now prove that sh eaf cohomology can be computed using hyper­ co v erin g s.

F o r sch e m e s, this theorem was first proved by J . - L . Verdier

in [7], V .7 .4 .1 . THEOREM 3 .8 .

L et X.

b e a s im p licia l s c h e m e .

T h en there is a natural

isom orphism of 8 -functors on A bSh(X.) H * (X .,

)

w here the colim it is in d ex e d by U ..

colim H *( (U ..)) in H R (X .).

P roof. A s seen in the proof of P rop osition 3 .7 , for any F e A bSh(X.) and any V .. in H R (X .), H °(X . , F ) - = .H ° ( V ..f F ) = H ° ( F (V ..)) » colim H ° (F (U ..)); m oreover, for any in jectiv e

I e A bSh(X .) and any

V ..

in H R (X .),

H * (X ., I)-==»H*(V.., I ) ^ H * ( I ( V . . ) ) ^ c o l i m H *(I(U ..)). T hus, it su ffice s

28

E T A L E HOMOTOPY OF SIM P LIC IA L SCHEMES

to prove that a short e x a c t se q u en ce

0 -»

-» F 2 -» F 3 -» 0 in A bSh(X.)

determ ines a long e x a c t seq u en ce ••• Let

colim H ^ F ^ U ..) ) -» colim H *(F 2 (U ..)) -> colim H *(F 3(U ..))

Gj = cokerC Fj->F2) a s p resh eav es on E t(X .) , and let G3 =

coke^G-^ ->F3) ,

so

that 0 ->

of p resh eaves on E t ( X .) .

-> F 3 -» G3 -» 0 is a short e x a c t seq u en ce

F o r each

U. . ,

0 -> F 1(U ..) -» F 2(U ..) -> G i(U ..) -» 0 ,

0 -» GjCU..) -> F 3(U ..) -> G3(U ..) -> 0 are e x a c t, thus yielding long e x a c t seq u en ces in cohom ology.

B ecau se

H R (X .) is left filtering and b e ca u se

th ese long e x a c t seq u en ces are functorial on H R (X .),

we conclude that it

suffices to prove that colim H *(G 3(U ..)) = 0 . L e t a e G3(U n n) rep resen t a given cohomology c la s s Hn(G 3( U ..) ) ,

some

n>0,

and U ..

a in

in H R (X .). B e c a u s e the sh eaf

a s s o c ia te d to the p resheaf G 3 is the zero sh e a f, we may ch o o se an e tale su rje ctiv e map W -> Un n su ch that a claim that tion 1 .7 ).

T

Un

n‘ (W) = V. -> Un

re s tric ts to

0 in G3(W ). We

is a s p e c ia l map over X n (s e e P ro p o si­

Nam ely, the appropriate su rje ctiv ity in dimension t > 0 is

equivalent to the lifting of geom etric points, which is equivalent to lifting maps from S p e ck ® A [ t ] . T his can be readily ch ecked using the definition of r ^ * ( C onsequently,

) and the ad jo in tn ess of cosk^ ^ ( ) and s kt l ( ) .

U '. = r ^ * (V .)

th at the fibre product U7. of X .

r * * ( U n ) is a s p e c ia l map over X . , so

of U.'. -» r ^ * ( U n ) 0.

T h en there is a

natural isom orphism of 8 -functors on A bSh(X .) H *(X . , ) ^ P roof. ings

H *(X . , ) .

The in clu sion of the homotopy ca te g o ry of C ech nerves of co v e r-

V. -> X .

U .. ^ X .

into the homotopy categ o ry

H R (X .) of hypercoverings

induces a natural transform ation H * (X .,

colim H *( (U ..)) = H * (X ., F e A b S h (X .),

) = colim H *( (Nx (V.)))

) of 5-fu n ctors on A b S h (X .).

F o r any

th is map is induced by a map of filtered co m p lexes, and

thus may be identified with the map on abutments of a map of sp e ctra l seq u en ce F y g .F\g ) = > Hs + t(X . ,F ) E .M f ’1 = = h ifV yX

E f * 1 = H ^ X S’ g .FS ) = > Hs + t(X . ,F ) where the first sp e c tra l seq u en ce is that of Prop osition 3 .2 and the 'E j- te r m is identified using the isomorphism

H *(X g ,F s )

colim H *(F g (Us )) provided by the left finality of H R (X .) -> H R (X g) . The corollary now follow s by applying A rtin ’s theorem to conclude that this map of sp e ctra l seq u en ces is an isomorphism. ■ F o r the sak e of co m p leten ess, we determine the e ffe ct of colim H*( (U ..)) on ab elian p resh eav es with the aid of Theorem 3 .8 . C O R O L L A R Y 3 .1 0 . (ab. g rp s.) n P (U j) ifl

L et X.

be a s im p licia l s c h e m e , and let P : E t ( X .) ° h

b e an a b elia n p re s h e a f with the property that P ( II U -) = lei

for any

II Uj -> X n in E t ( X . ) . iel

d en o te the s h e a f a s s o c ia t e d to P natural isom orphism

L et

P # : E t ( X .) ° - (ab. g rp s .)

(c f. [5 9 ], II.2 .1 1 ).

T h en there is a

30

E T A L E HOMOTOPY O F SIM P LIC IA L SCHEMES

H *(X . , P #) — » colim H * (P (U ..)) w here the colim it is in d e x ed by U .. -» X .

in H R (X .) .

P ro o f. By Theorem 3 .8 , it su ffice s to prove that the natural map P -> P # determ ines an isomorphism of ch ain com p lexes colim P (U ..) -> colim P #(U ..) . B ecau se

P

and P # have isom orphic s ta lk s at every geom etric point and

b e ca u se

P

“ com m utes” with d isjoin t unions, we conclude the following:

If a 6 P (U S t ) goes to 0 £ P #(U g t ) , then there e x is ts an e ta le s u rje ctiv e map U '-> U g j. su ch that a re s tric ts to

0 £P(U0;

and if /3 £ P #(Ug

then there e x is ts an e ta le su rje ctiv e map U"-> U tion of (3 in P #(U ")

f su ch that the re s tric S, I is in the image of P ( U " ) . As argued in the proof

of Theorem 3 .8 , given any e ta le s u rje ctiv e map W -> U S, fL there e x is ts a map of hypercoverings

U.'. ^ U . .

W -> Ug t . C onsequently, phism for any s ,t > 0

,

such that Ug ^

Ug ^ fa cto rs through

colim P (U g t ) -> colim P #(U g p

as required. ■

is an isom or­

4.

E T A L E T OPOLOGICA L T Y P E

As we observed in the la s t ch ap ter, the sh eaf cohomology of a sim p licial sch em e

X.

is determined by its h ypercoverings.

top olog ical type of X . , ( X .) e ^,

The e ta le

is e s s e n tia lly the in verse sy stem of

sim p licial s e ts given by applying the con nected component functor to the h ypercoverin gs.

As we s e e in Chapter 5, the sh eaf cohomology of X.

with lo cally co n stan t c o e fficie n ts can be computed from the homotopy type of ( X .)e t .

\/ Our use of general hypercoverings rather than only C ech nerves does

com p licate our co n stru ctio n , although this com plication is primarily one of notation.

The reader is urged to con sid er only C ech nerves when in itially

con sid erin g the e ta le to p o lo g ical type:

the in verse system of sim p licial

s e ts a s s o c ia te d to C ech nerves (the C ech to p o lo g ica l type) is shown in P rop osition 8 .2 to be weakly equivalent to ( X .) et in most c a s e s of in terest.

On the other hand, there are s e v e ra l important ad vantages of our

definition of the e ta le to p ological type using general h ypercoverings. Namely,

( X .) e ^ has the “ c o r r e c t” weak homotopy type for all lo cally

noetherian sim p licial sch em es and ( )et

is indeed a refinement of Artin-

M azur’s e ta le homotopy type (P ro p o sitio n 4 .5 ).

P erh aps most important,

our con stru ction applies to other s ite s (e .g ., the Z arisk i s ite ) in which \/ C ech cohomology differs from derived functor cohom ology. T his co n stru c\/ tion of the e ta le to p o lo g ical type is based on that of the C ech to p o lo g ical type first appearing in [34] and [51].

The e s s e n tia l idea, su ggested by

work of S. Lubkin [5 4 ], is to “ rig id ify ” cov erin g s by providing e ach co n ­ nected component with a distinguished geom etric point. B e c a u s e the e ta le to p o lo g ical type is not a sin g le sim p licial s e t (or s p a c e ), e ta le homotopy theory is somewhat off-putting at first g lan ce. 31

As

32

E T A L E HOMOTOPY OF SIM P LIC IA L SCHEMES

defined in Definition 4 .4 ,

( X .) e t is a pro-sim p licial s e t (a functor from a

left filtering categ o ry to sim p licial s e ts ) .

Although pro-ob jects were first

sy ste m a tica lly employed in e ta le homotopy (c f. [8], Appendix), the s p e c ia l c a s e of pro-groups has been widely used in g alo is cohomology [66],

At

first acq u ain tan ce with su ch p ro -o b jects, one is tempted to take an in­ v erse lim it.

Not only does one lo se stru ctu re once the in verse limit is

applied, but a lso one obtains in co rre ct in v arian ts:

the cohomology of a

p ro-sim p licial s e t is a kind of continuous cohomology of its in verse lim it. The know ledgeable reader will reco g n ize the c lo s e analogy of our c o n ­ stru ctio n of ( X .) e j. to shape theory in topology [28]. sid ers a “ n ic e ” to p o lo g ical s p a c e

T

In fa c t, if one co n ­

(e .g ., a finite C.W. com plex) and a

“ h ypercoverin g” of T with each component a co n tractib le open su b se t of T ,

then the sim p licia l s e t of con n ected com ponents of this hyper­

covering has the “ s a m e ” homotopy type a s

X

([8 ], 1 2 .1 ).

We begin with the following proposition which provides the rigidity of our con stru ction .

The to p o lo g ical analogue of P rop osition 4 .1 is the fa ct

that a map of con nected covering s p a c e s

X " of a s p a c e

X

is

determined by its value on a sin g le point. PROPOSITION 4 .1 . and V -» X

be a sc h e m e ,

eta le a n d sep a ra ted .

equ a l if f °u = g °u P roof.

L et X

U-^X

eta le with U c o n n e c te d ,

T h en two maps f,g : U -» V

g a s s e c tio n s of the projection

Such se c tio n s are open ([5 9 ], 1 .3 .1 2 ) and clo se d (b ecau se B ecau se

a re

for som e g eo m etric point u : Speck -» U .

We interpret f and

sep arated ).

over X

V

U x V -> U . X X

is

U is co n n ected , we thus may identify a se ctio n

with a ch o ice of con nected com ponents of U x V isom orphic to U . X B ecau se

u and f ° u

(re sp e ctiv e ly ,

g ° v ) determ ine a geom etric point of

the connected component of U x V corresponding to f (re s p e c tiv e ly , X f equals

g whenever f ° u

equals

g°v. ■

g ),

4. E T A L E TO PO LO G IC A L T Y P E

33

The reader should bew are th at P ro p o sitio n 4 .1 is fa ls e if one merely assu m es that f and take

g agree on a sch em e-th eo retic point.

X = Spec k and U = V = Spec K with

K /k

F o r exam ple,

a finite g alo is exten sio n ;

then each d istin ct automorphism of K over k determ ines a d istin ct map from U to

V , but th e se d istin ct maps of co u rse ag ree on the unique

schem e th eo retic point of U . We next introduce the definition of a rigid (e ta le ) covering of a schem e X . B e c a u s e we sh all employ the uniqueness property of Prop osition 4 .1 , we index the con n ected com ponents by the geom etric points of X and provide each con nected component with a distingu ish ed geom etric point. A first problem a ris e s in that an e ta le open U of X joint union of its co n nected com ponents.

may not be a d is ­

A secon d problem is the

aw kw ardness th at the co lle c tio n of all geom etric points of X We re ca ll th at a sch em e X

is not a s e t.

is said to be lo ca lly n o etherian provided

that it is a union of affine noetherian sch em es ( i .e ., sp e ctra of noetherian rings).

B e c a u s e the Z a risk i to p o lo g ical s p a ce of a lo cally noetherian

schem e has the property th at every point has a sy stem of neighborhoods in which each d escen ding chain of clo se d s u b s e ts is fin ite, we readily conclude th at each sch em e-th eo retic point of a lo ca lly noetherian schem e has su ch a neighborhood which is a ls o irreducible. th at the con n ected com ponents of X

T his e a sily im plies

are open as w ell as clo se d ([4 7 ],

I. 6 .1 .9 ) . To avoid s e t th eo retic problems, we ch o o se for e a ch c h a ra c te ris tic p > 0 an alg e b ra ica lly clo se d field

12p which is su fficien tly large to

con tain su bfields isom orphic to the residu e fields of c h a ra c te ris tic every sch em e we co n sid er.

p of

(At this point, s e t th e o rists would su g g e st we

fix a “ u n iv e rse ” and con sid er only th ose sim p licial sch em es which in each dimension lie in th is u n iv e rse .) Once su ch a ch o ice is made, we define the s e t of g eo m etric points of X , denoted geom etric points

x : Spec 12 -> X

X , to c o n s is t of all

where 12 = 12p with

p the residue

c h a ra c te ris tic of the image schem e th eo retic point of X .

34

E T A L E HOMOTOPY OF SIM P LIC IA L SCHEMES

With th e se p relim in aries, we introduce rigid co v erin g s. DEFINITION 4 .2 . scheme

X

A rig id co v erin g a : U -> X

of a locally noetherian

is a disjoint union of pointed e ta le , separated maps ax : Ux ' ux

where each

Ux

Vx € X

X ’x '

is co n nected and ax °u x = x : Spec 12 -»X . A map of

rig id co v erin g s over a map f : X -> Y

of sch e m e s,

cf> : (a : U -»X) ->

(j8 : V -»Y) is a map cf): U -» V over f su ch that 0 ° ux = vf(x ) f ° r x e X . If U -> X and

V -> X are rigid coverings of X

and Y

over a

R

third sch em e

S , then the rig id product U x V - ^ X x Y S

the open and clo se d immersion of U x V ^ X x Y S

union indexed by geom etric points

is defined to be

S

given a s the d isjoint

S

x x y of X x Y

of

S

ax x

where

(U

:

(ux

x Vy)0 , ux x vy

X x Y , xxy

x V ) 0 is the con n ected component of Ux x V

s y

s y

containing

ux x vy • ■ Of co u rse, P rop osition 4.1 implies th at th ere is at most one map between rigid co v e rin g s.

T his im plies that R C (X .),

rig id co v erin g s of a sim p licial sch em e

the ca tego ry of

X . , is a left directed categ o ry .

We proceed to define rigid hypercoverings in su ch a way that they too determ ine a left directed categ o ry .' PROPOSITION 4 .3 .

L et X.

b e a lo ca lly n o eth eria n sim p licia l s c h em e .

A rig id h y p erco v erin g U .. -» X .

is a h y p erco v erin g with the property that

Us ,t - (c o s k j® Us )t is a rig id co v erin g for e a ch s ,t > 0 s u c h that any map a : A [s '] -> A [s]

4. E T A L E TO PO LO G IC A L T Y P E

35

in d u ces a map of rig id c o v erin g s over a : (cosk^._® Ug for ea ch t > 0 .

A map of rig id hy p er c o v erin g s over a map f : X . -» Y .

a map of b isim p licia l s c h e m e s 0s t :U st-V s t

cf): U .. -> V ..

is

over f su c h that

X is a map of rig id c o v e rin g s over c o s k t_ 10 : ( c o s k ^ U g )t

Y -» (c o s k ^ jV g )^. for ea ch s ,t > 0 . of X . , H R R (X .),

T h e ca teg o ry of rig id hy p er c o v erin g s

is & left d ire c te d ca tego ry .

P roof. B e c a u s e any sim p licial map cf>s : Ug the

-> ( c o s k ^ Ug ')j.

(t-l)-tru n c a tio n of

determ ines

Vg

has the property that

co sk t l 0 s

, we conclude by

Prop osition 4 .1 th at there is at most one map betw een any two rigid hyper­ coverings over a given map of sim p licial sch e m e s.

To prove H R R (X .)

left d irected , it thus su ffice s to observe that if U .. -> X .

is

and V .. -> X .

R are rigid hypercoverings then their rig id product U .. x V .. -> X . is a rigid X. R R hypercovering mapping to both, where (U .. x V ..)g t -» (co sk t l (Us x Vs )T X. ’ • Xs is defined to be the re strictio n to

(c o s k t l (Ug

R x Vg ))t C ‘ xs

(c o s k t l Us )t x (co sk t l Vs )t of the rigid product over X g of xs Us ,t - (c o s k t - l Us .) t

and Vs ,t - (c o s k t - i Vs . ) f ■

We remind the reader that a p ro -ob ject of a categ o ry from some sm all left filtering category to p ro-object F : I -» C by {F- ; i e l i

of p ro -ob jects from F : I -> C to lim colim Homc ( F ( i), G (j) ) ; J

C . We sh a ll often denote a

or simply

ignore the s e t th eo retic requirem ent that

C is a functor

iF^i,

and we sh all usually

I be a sm all categ o ry .

A map

G : J -> C is an elem ent in the s e t

in other words, a map from F

to G is a

I

com patible c o lle c tio n of elem ents (indexed by J ) in colim Homc ( F ( i ) , G ( j)), each of which is determined by a map

36

E T A L E HOMOTOPY OF SIM P LIC IA L SCHEMES

F (i ) -> G(j) for some from F : I

C to

A strict map of p ro -o b jects

i (depending on j e J ).

C is a pair co n sistin g of a functor a : J -» I

G:J

: F °a -> G . Of co u rse , a s tr ic t map of

and a natural transform ation

p ro-ob jects determ ines a map of p ro -o b jects. If F : I -> C is a pro-object and if a : J -» I is a functor between left filtering c a te g o r ie s , then (a : J -* I , id : F °a -> F ° a ) F

to F o g .

is a s tr ic t map from

If the functor a : J -» I is left final, then we readily con clu d e

the e x is te n c e of a (not n e c e s s a rily s tr ic t) map of p ro-ob jects inverse to th is ca n o n ica l map.

F °a -> F

In other w ords, the s tr ic t map F -» F °a

is

an isomorphism of p ro -o b jects, even though it may not have a s tr ic t in verse. \/ * The C e c h top o logica l type of X . is defined to be the p ro-sim plicial s e t A (X .)ret = 77ANx ing a rigid covering

: R C (X .) -> ( s . s e t s ) (notation is that of [34]) sen d ­

U .. -» X .

to the sim p licial s e t

77(A(NX (U .)))

in dimension t a s the s e t of con n ected com ponents of the

given

t-fold fibre

product of Uj. over X j.. T his definition su g g e s ts our definition of the e ta le to p ological type. DEFINITION 4 .4 .

Let

X.

be a locally noetherian sim plicial scheme.

The eta le topologica l type of X . sim p licial s e t

is defined to be the following pro-

( X .) et - 77° A : H R R (X .)

sending a hypercovering U ..

of X .

(s . s e ts )

to the sim p licial s e t

con nected components of the diagonal of U .. s e t of co n n ected com ponents of Un

).

(s o that

If f : X . -> Y .

7t(AU..) of

77(AU..)n is the is a map of lo ca lly

noetherian sim p licial sch e m e s, then the stric t e ta le to p o logica l type of f is the s tr ic t map

fe t : ( X . ) e t , ( Y . ) et

given by the functor f* : H R R (Y .) -» H R R (X .) and the natural transform ation ( X .) e t ° f * -> ( Y .) et induced by the natural maps in H R R (Y .).

f * (V ..) -> V ..

The e ta le top o lo g ical type functor ( ) e j .:( l o c . noeth. s . sch em es) -> (p ro -s. s e ts )

for V .. -> Y .

4. E T A L E T O PO LO G IC A L T Y P E

sen ds a lo cally noetherian sim p licial schem e f : X . -> Y .

X.

37

to ( X .) e {- and a map

to the map of pro-sim p licial s e ts determined by f ^ . ■

In the above definition, the functor f* : H R R (Y .) ^ H R R (X .),

the

rig id pu ll-ba ck functor, is defined in term s of the pull-backs of rigid co v erin g s.

If V -> Y

f*(V -»Y) = U -> X

is a rigid covering and f : X -> Y

is the disjoint union of pointed maps

(Vf(X ) x X ) 0 , f(x) x x ^ X ,x component of Vf. . x X Y V.. -♦ Y.

is a map, then

where

(V £ ^ x X ) Q is the con nected

con tain in g the geom etric point f(x) x x . If

is a hypercovering of Y. , then f*(V .. -^Y.) = U .. -> X .

is

defined for any s ,t > 0 by Us ,t - ( c o s k ^ U s A If X .

= f*(VSft - ( c o s k ^ V g .) ) •

is a sim p licial sch em e provided with a ch o sen geom etric point

x : Spec Q

then

X.

(or, more p re cise ly

po in ted sim p licia l s c h e m e .

C learly , if X .

(X . , x ) ) is said to be a

is a pointed lo cally neotherian

sim p licial sch em e, then ( X .) e £ is naturally a p ro-object in the categ o ry (s . s e ts ^ ) of pointed sim p licial s e t s .

M oreover, if f : X . -> Y .

is a

pointed map of pointed lo cally noetherian sim p licial sch e m e s, then f ^ is naturally a s tr ic t map of p ro-(s. s e t s ^ ) . O riginally, the e ta le homotopy type of a sch em e X ht = 7 7 °A :H R (X ) - » H,

X was defined to be

where H is the homotopy categ o ry of sim p licial

s e ts defined by inverting weak e q u iv alen ces ([8 ], 9 .6 ). PROPOSITION 4 .5 .

L et X

(X ® A [0 ])e £ e pro-(s. s e ts ) X ^ € Pro_W

be a lo ca lly n o eth eria n s c h e m e , let

be d e fin e d a s in D efin itio n 4 .4 , an d let

d e fin e d a s a b o v e . If (X ® A [0 ])e £ is v iew ed in pro-K

applying the fo rg etfu l functor (s . s e t s ) - >K, ph ic to X ^

by

then (XA[0])e ^ is isom or­

in pro-K .

P ro o f. We readily verify th at a rigid hypercovering U .. -* X ® A [0 ] is of the form (U. -> X )® A [0 ],

where U. -» X

is a rigid hypercovering of X

38

E T A L E HOMOTOPY OF SIM P LIC IA L SCHEMES

(i .e .,

Un -> (co sk n l U .)n is a rigid coverin g for n > 0 ).

T hus, it

su ffice s to verify that H R R (X) -> H R (X ) is left final, where H R R (X ) is the left d irected categ o ry of rigid hypercoverings of X . If g : U -» V is a s u rje ctiv e map, then a “ c h o ice of right inverse to g* ” is a function g” 1 : V -> U right in verse to g^ : U -> V sending u : Spec 12 -> U to

g *(u ) = u ° g : Spec 12 -> V . If g : U

and s u rje ctiv e , and if g~*

V is e tale

is a ch o ic e of right in verse to g^ , we let

= J J (U y .g ^ C v ) ^ v ,v ) v fV

be a rigid covering of V , open of U containing

where Uy is a co n n ected , sep arated Z a risk i

g ~ * (v ).

C learly ,

R ( g , g ^ ) -►V fa cto rs naturally

through g : U -> V . U sing this co n stru ctio n , we exh ib it for U. -> X in H R R (X ) naturally mapping to U. -> X . Define R (g 0,g o * ) -* ^ , where

g Q: U Q -> X

a ch o ice of right in verse to

g 0* .

in HR(X) , R (U .) -» X R (U .)0 -> X

to be

is the augm entation map and g ^

is

Inductively, define R (U .)n ->

(c o s k n l R (U .))n to be R (g n, g ^ ) -> ( c o s k ^ R C U .) ^

where g n is the

projection of the fibre product of (c o s k n l R (U .))n -» (c o s k n l U .)n «- Un onto

(co sk n l R (U .))n and g“ * is induced by the degeneracy maps on

(s k n_ 1R (U .))n C (c o s k fl_ 1R (U .))n . T o com plete the proof of left fin ality , let sim p licial sch em es with U. h : H. -> U. 3 .4 .

is s p e c ia l, we co n stru ct a map

in H R (X .) such that h ° f : W. -> U.

is in H R R (X) .

f 0 : W0 ^HQ to be R (s,6~1) ^ H q ,

augm entation map and

e~ l

s a tis f ie s

b;+.°E ~ 1(x )

the distinguished geom etric point above

Uq -> X ).

in H R (X ), and let

be the left equalizer con stru cted in the proof of Proposition

We define

ux

in H R R (X ) and V.

Using the fa c t that H. -> U.

f : W. -> H.

U. 3 V. be two maps of

With e " 1 so ch o sen ,

x

where =

e:Hq^X

ux for all x

f

is the

X

(with

of the rigid covering

h Q° f 0 : WQ -» U Q is a map of rigid c o v e r­

ings of X . Inductively, we let g^ : Kn

L n denote the map induced by

39

4. E T A L E TOPOLOGICAL T Y P E

h from the fibre product Kn of Hn -> (c o s k n l H .)n (c o s k n l U .)n ,q = colim H ^(F(V V..

)) = > colim HP+C1 (F (U ..))

p*

u..

)) => colim HP+C1 (F (V ..)) V..

We show th at this map is an isomorphism.

By P ro p o sitio n s 3 .4 and 4 .5 , it

su ffices to prove th at the re strictio n map H R R (X .) -» H R R (X p ) is left final for each R r * - ( W .) ^ X . P settin g

p > 0.

Imitating the proof of P ro p o sitio n 3 .4 , we define

in H R R (X .) a s s o c ia te d to

W. - X n in H RR(X ) by P P

( R r ^ ( W .) ) c to be the rigid product of the rigid pull-backs of P s. W. -» X via the maps X -> X n indexed by A [ s ]_ . Then ( R r ^ ‘(W .)) -> P b P P P P* Xp facto rs through W. -» Xp so that H R R (X .) HRR(Xp) is left final. ■

40

E T A L E HOMOTOPY OF SIM P LIC IA L SCHEMES

The proof of the following proposition makes frequent use of the uniqueness of maps betw een rigid hypercoverings to insure the validity of sim p licial id en titie s.

The u sefu ln ess of th is proposition will become

apparent when we co n sid er function co m p lexes. PROPOSITION 4 . 7 .

L e t X.

be a lo ca lly n o eth eria n sim p licia l sc h e m e

and let n b e a p o sitiv e in teg er. (X .® A [n])e t

T h en the natural strict map

( X .) et x A[n]

in

pro-(s. s e ts )

is an isom orphism with in v ers e pro v ided by a stric t map ( X . ) ^ x A[n] (X . ® A [n])e t . P roof. B e c a u s e the co sk e le to n functor commutes with d isjoint unions, U .. ® A[n] -» X . ® A[n] defined by (U.. A [n]) S j .L = Ub ^I . ®A[n] b is a rigid hypercovering of X .® A [n ] whenever U .. is a rigid hypercovering of X . . Therefo re, the id en tification of ( X ,) e ^.®A[n] with (X . ® A [n])e ^. ° ( ? ® A[n]) determ ines the s tr ic t map (X .® A [n ]) ^ -» ( X .) ^ x A[ n ] .

To prove the

proposition, it su ffice s to exhibit a right adjoint for ( ? ® A [n ]), p : H R R (X.® A[n]) -> H R R (X .),

so that the adjunction transform ation

( ? ® A [ n ] )° p -> id determ ines a stric t map from (X . ® A [n])ej. ° ( ?® A [n]) = ( X .) ej.xA [n ] to (X .® A [n ])e ^. and (? ® A [ n ] ) is left final. For U.'. -> X .

U .. -> X .® A [n ] in H R R (X. ® A[n]) , we define p(U .. - X . ® A[n]) = in H R R (X .) by se ttin g

Uk

equal to the rigid product in

n+k H R R(X^) indexed by the ( k ) non-degenerate sim p lices x A [n])n+k of

**(U £ + n^ X n+k® {a '| ) in H R R (X k) , where

is th at copy of X k+n® A [n]k+n indexed by

< o, o '> e (A[k] X k+n® { a '}

o ' and where Uk+n is the

restrictio n of Uk+n -> (X . ® A [n])k+n to X k+n® {c r 'i.

If a : iO,---,kS

! 0, ---, mS is a n on-decreasing map, we define the com posite Uk. " a * (Uk+n.) by « /oP V : U m. ^ ^ (U ra+n.) ^ ff^ Uk + n .)'

pr o a : ! ^

->

where

< p , p > e (A [m ]x A [n])m+n and a\ l0,---,k+n| -> { 0 , - - -, m+n! are defined as follow s:

define a ' by a'Q ) = a '( j - 1 ) + 1 for j su ch that

a ( j ) = cr(j—1)

4. E T A L E TOPOLOGICAL T Y P E

and a '( j ) =a X j - 1 ) + a ( a ( j ) ) - a ( a ( j - l ) )

if a ( j )

41

^ a (j-l) ;

define

p: l0,***,m+n! -> {(),•••,mi to be the unique su rje ctiv e , non-decreasing function such that a ' ( j)

=

i and

a (j)

( i { i)

whenever there e x is t s a

=

a ( j + l ) . We readily verify that

=

unique su rjectiv e non-decreasing maps fitting

A|n]

fi

j su ch that

and

are the

\l

inthe commutative squares

------ ^ ----- A[k+n] -------- - ----- ►A [k ]

a

id

a

A[n] - ------ ^ ------ A[m+n] ------- £------►A[m]

B e c a u s e th ese squares commute, a*(U ^+n) covering

a ': Um+n -> Uk+n

a :X m ^X^.

Moreover, if

r e s t ric ts to /**(Uj£+n ) : !0,--*,m i -> {0, •**,p i is

another non-decreasing map, we readily ch eck that (/3 ° a ) ' = fact that U.'.

The

is a well-defined bi-simplicial scheme now follows using

rigidity. We briefly sk e tch the proof of the adjoin tness of ?® A [n ] and p . f :(W ..® A [n ] - * X .® A [ n ] ) -* (U ..-> X . ® A[n])

If

is a map in HR R(X.® A[n]) ,

then we obtain (W.. ->X.) -> ( p (U ..)-» X .) in H R R (X .) by defining \ p(U..)k . ^ a * (U k+n.) as the restriction °f a*(Wk ) .

^k+n: \ + n .® ^ a ^

Uk+n.

to

C onversely, if g : (W.. ^ X . ) -» (p (U ..)-+ X .) is a map in H R R ( X .) ,

then we obtain (W.. ® A[n] -» X . ® A[n]) -> (U .. -^X. ® A[n]) whose restriction to Wk ® l s l ^ U ^

is defined by ) ->

Uk , where < o ,o '> ^ is some non-degenerate simplex of (A[k] x A[n])k+n and where 8 : cr*(U^+n )

Ufc

is induced by 8 : Uk+n -> Uk

with

S : {0,***,k| -» { 0 , * “ ,k+ni any strictly increasing map such that and 8 : A[n]k+n -> A[n]k sends

o ' to

o ° 8 = id

e . ■

The following is an immediate corollary of Proposition 4 . 7 . C O R O L L A R Y 4 .8.

L e t f, g : X. -> Y .

b e two maps of lo ca lly noetherian

sim p licia l s c h e m e s re la ted by a sim p licia l homotopy. d eterm in e the sam e map in pro-K. ■

T h en f e ^ and ge ^

5.

HOMOTOPY INVARIANTS

In th is ch ap ter, we co n sid er the homotopy groups and cohomology groups of the e ta le to p o lo g ical type ( X .) e £-

We identify

ttq

and

with their alg eb raic cou nterp arts (p artial resu lts about the higher homotopy groups are to be found in later ch a p te rs).

In p articular, our

study of the fundamental group requires a review of d e sce n t techniques in the co n text of principal G -fibrations.

U sing our id en tification of funda­

mental groups, we a ls o verify th at the cohomology of ( X .) e j. with abelian lo ca l co e fficie n ts is isom orphic to the cohomology of X . cie n ts in the corresponding lo ca lly co n stan t sh eaf.

with co e ffi­

We point out that

th ese homotopy and cohomology groups are invariants of the homotopy type of ( X .) e j_; by Theorem 3 .8 , the top ological type itse lf determ ines in some se n s e the cohom ology groups of X .

with valu es in any abelian

sh eaf. DEFINITION 5 .1 . sch em e.

For

Let

n > 0,

X . ,x

be a lo ca lly noetherian pointed sim p licial

we define the pro-object of pointed s e ts

* n((X . >XW

= nn ° ( x * »x )e t : H R R (X .) -> ( s e t s ) .

T hu s, for n > 1 , 77fl((X . ,x )e t) is a pro-group; for n > 2 , 7rn((X . ,x )e t) is a pro-abelian group. 7t ° A (U ..) for some 5 .8 ).

F o r any

L e t M be an ab elian lo ca l co e fficie n t system on

U. . eHRR( X. )

n>0,

(s e e d iscu ssio n preceding Corollary

we define

Hn(( X .)e t ,M) = colim Hn(7 7 °A (U :.),j*M )

where the colim it is indexed by j : U.'. ->U ..

42

in H R R (X .)/U .. . ■

43

5. HOMOTOPY INVARIANTS

The reason we co n sid er the homotopy pro-groups rather than their in verse lim its is th at too much information is lo st upon applying the in verse limit functor (which is not even e x a c t, unlike the d irect limit functor colim ).

As we se e in the next proposition,

770((X . ,x ))

is actu ally

(isom orphic to) a pointed s e t. PROPOSITION 5 .2 .

L e t X . ,x

sch e m e and let t7q(X .

,x )

b e a lo ca lly n o eth eria n p o in ted sim p licia l

d en o te the p o in ted s e t of c o n n e c te d com ponents

of the p oin ted sim p licia l s e t n(X. , x ) , 77q(X. the po in ted pro-set 77Q((X . ,x )e t)

,x )

= tTq(7t{X. , x ) ) .

is isom orphic in p ro -(sets^ )

T h en

to the

pointed s e t ?7q(X. ,x ) 770((X . ,x )e t) Furtherm ore, X .

ac 77q(X . , x) .

is a d isjo in t union of non-trivial sim p licia l s c h e m e s

co rresp o n d in g to e lem en ts a e 77Q(X . ,x )

with no X ?

X?

e x p r e s s ib le a s a

non-trivial d isjo in t union. P roof. We ignore the b ase point and prove that the natural map 77Q( ( X .) e t)

77q(X.) is an isomorphism of p ro -se ts. F o r th is, it su ffice s to prove that this map induces a b ijectio n Hom(77Q(X .),S ) -> Hom(770( ( X .) e t ,S) = colim Hom(770(77(AU..)),S)

for any s e t

S . To prove th is, it su ffice s to prove that the natural map

H °(X . ,S) = Hom(770(X .),S ) -> Hom(n-0(n (A U ..)),S ) = H °(U .. ,S)

is a b ijection for any s e t

S . Viewing S a s a co n sta n t sh eaf of s e ts , we

may apply the proof of P rop osition 3 .7 to verify this la s t b ijectio n .

The

secon d a sse rtio n of the proposition is verified by in sp ection . ■ G rothendieck’s alg e b ra ic interpretation of fundamental groups is given in terms of coverin g s p a c e s .

T h o se con nected covering s p a c e s which

correspond to normal subgroups of the fundamental group are principal G -fibrations, where

G is the quotient group.

44

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

DEFINITION 5 . 3 . (d is cr e te ) group. sch em es

Let

X.

be a sirnplicial scheme and let

A p r i n c ip a l G - fib r a tio n over X .

f:X .'n > X .

G be a

is a map of sirnplicial

together with a right actio n of G on X.' over

X.

su ch that a .)

T here e x is ts an e ta le , s u rje ctiv e map U ->Xq and an isomorphism

of sch em es

U x Xq-+U®G

over U commuting with the actio n of G

xo (w here G a c ts on U®G b .)

by right m u ltiplication).

F o r each map a : A[n] -» A[m] in A ,

\ r '

O'

v

m

fn

x m ------

Xn

A map of principal G-fibrations over X . , X ." - > X ',

isomorphism of sim p licial sch em es over X . G.

'

n

fm

is ca rte s ia n .

the square

commuting with the actio n of

We denote the categ o ry of principal G-fibrations over X . In the following lemma, we re-in terp ret the above categ o ry

in terms of the categ o ry

is an

n ( X 0, G) plus “ d e sce n t d a t a .”

by Ei(X. ,G) . II(X. ,G)

We intend the

following d iscu ssio n to be a first introduction to the technique of d e sce n t. L E M M A 5 .4 .

L et X.

the ca tego ry II(X. ,G )

b e a sim p licia l sc h e m e and let G be a group. T h en of p rin cip a l G -fibrations over X .

the ca tego ry < II(X q , G) ; d.data > d e fin e d a s fo llo w s.

is eq u iv a len t to

A n o b ject of

< II(X q , G ) ; d .d ata > is a prin cip a l G -fibration over X Q, q Q: X Q ^ X 0 , togeth er with “ d e s c e n t d a ta ” —nam ely, an isom orphism in I1(X 1,G )

sa tisfy in g d* = d^°dQ

(q0 : X q ^ X 0 ; 0 )

to ( r 0 : X g - > X 0 ;^r)

in II(X 2 , G ) .

-» d ^ q ^ A map from

is a map 0 : X q - > X q

sa tisfy in g the condition i/r ° d * 0 = d * (9 ° 0

in I ^ X p G ) .

in II(X 0, G)

45

5. HOMOTOPY INVARIANTS

P roof.

T o a principal G -fibration q : X. ' - > X .

over X . , we a s s o c ia te

the principal G-fibration q 0 : X Q - > X 0 over X q together with the isom or­ phism 0

given a s the com posite

q1

dQ(qQ)

d * (q Q) . C learly ,

q h> < q q, 0 > determ ines a faithful functor from II(X . ,G) to

< n ( X 0,G); d. data > . C on versely, let q Q: X Q ^ X 0 and

: dQ(qQ) -> d * (q Q) determ ine an

elem ent of < I I ( X 0 ,G ); d. data > . We define

q: X. ' - >X.

by settin g

X^

equal to the fibre product of d Q° ••• ° d Q: X n -> X Q and q 0 : X Q ^ X Q and by settin g

qn : X^ -* X n equal to the projection .

Define s - : X ^ _ j -» X^

to be s j; x id a Y 'q ; define d-i : nX ' -> X n— ' i 1 to be ld;x i dY / aq and define

dfl : X^ -> X ^ _ j

for 0 X .

is a well-defined map of sim p licial sch e m e s. More­

over, using th is e x p licit co n stru ctio n ,

we s e e that if (q 0, 0 ) and ( r Q, if/)

are elem ents of < I I ( X q,G ); d. d a t a > ,

then a map 0 : q Q ^ r Qof sch em es

over X q with ifrod*d - d * 0 ° 0 q h> r .

determ ines a map of sim p licial sch em es

In p articu lar, the G -action on q Q determ ines a G -action on q ,

so that q : XT-> X .

is an elem ent of II(X. ,G ).

The sam e argument

implies th at a map (q Q, 0 ) -» (r Q, t/r) in < II(X q ,G ); d. data > determ ines a map q

r

in II(X . ,G ) . ■

F o r Y .. is a b i-sim p licial

sch em e, we interpret the category

II(A Y .. ,G ) in a sim ilar fashion. PROPOSITION 5 .5 . ( d is c r e t e ) group.

L e t Y ..

be a b i-sim p licia l s c h e m e an d G a

T h en ca tego ry II(A Y .. ,G)

d e fin e d a s follow s.

is eq u iv a len t to the ca tego ry

A n o b ject of

is a p rincipa l G -fibration q : Y.'-> Y Q p ro v ided with an isom orphism cj> : dQ(q)

d*(q)

a map (q, cjT) dition

in

II(Y1 ,G ) sa tisfy in g

(r, \jf) is a map 0 : q -» r

°dQ0 = d * 0 ° 0 .

d* = d* : d qY.' -> d*Y." in I^ Y j

is equivalent to a map 77(U0

),u

d * 0 = d ^ °dQ0 .

47

5. HOMOTOPY INVARIANTS

Since Ug

is a hypercovering of X g for s > 0 ,

the categ o ry of

principal G -fibrations over tt(Us ) is equivalent to the categ o ry of principal G -fibrations over ([8 ], 1 0 .7 ).

Xg

whose re strictio n to

Ug Q is trivial

The seco n d a sse rtio n of the proposition is now immediately

implied by Lemma 5 .4 .

To prove the first a s s e rtio n , it su ffice s to re ca ll

that Hom(7r1((X . ,x )e j.),G) = colim Hom(771(77(AU..),u),G ) by definition and that if X .'^ X .

is a principal G-fibration then there e x is ts

the property that X 'Q x U q 0 - >UQ 0 *s xo (if U - ^ X q triv ia liz e s the rigid covering

U .. -» X with

v ia l principal G-fibration

’

X q - » X q , define

U ..

to be the “ rigid n erv e” of

R F q *(U) -» X . ; cf. proof of C orollary 4 .6 ). ■

Prop osition 5 .6 e s ta b lis h e s the following useful fa ct which we s ta te a s a sep arate co ro llary . groups

We im plicitly u se the fa c t that a map of pro­

{ H - ; i e l i ^ l G j ; j e J i which induces a b ijectio n

Hom(lH^S,K) for a ll groups

Hom(lGj S,K)

K is n e ce ssa rily an isomorphism of pro­

groups. C O R O LL A R Y 5 .7 .

L et X. ,x

sim p licia l s c h e m e .

L et

C

be a p o in ted , c o n n e c te d , lo ca lly noetherian

be any left filterin g su b ca teg o ry of H R(X. ,x )

(th e homotopy ca teg o ry of po in ted hyper co v e rin g s of X . , x ) with the property that for any eta le s u rje c tiv e map U -> X Q th ere e x is t s som e U .. -> X .

in C s u c h that U Q Q fa cto rs through U - » Xq.

functo rs

C -> H R (X. ,x ) H q q , where v ® l : Spec 0 -> //o m (A [l],V Q ) Q is the co n stan t homotopy. If T . a functor

is a sim p licial s e t, then a lo ca l c o e ffic ie n t sy stem on T .

is

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

48

L : (A /T .)° assig n in g to each sim plex

(s e t s )

t m e Tm a s e t

A[m] in A an isomorphism

L (tm) and to ea ch

a : A [ n ] -»

L (a ) : L (a (tm)) -» L ( t n) (c f. [42] for a d isc u s -

sio n 'o f A / T . , the ca te g o ry of sim p lices of T . ).

A map of lo ca l co e ffi­

cien t sy stem s is a natural transform ation of fu n ctors. c la s s of lo ca l co e fficie n t sy stem s isom orphic to to S on the pointed, con nected sim p licial s e t

L

An isomorphism

with fibres isomorphic

T . , t naturally corresponds

to an eq uivalen ce c la s s of homomorphisms n 1(T . ,t) - Aut(S) , where the eq u ivalen ce relation is generated by inner automorphisms of A u t(S ). Such an eq u ivalen ce c l a s s of homomorphisms naturally corresponds to an isomorphism c la s s of principal A ut(S)-fibrations on T . . For

{T? ; i e l! e pro-(s. s e ts ) , a lo c a l co e fficie n t system on !tM

eq u ivalen ce c l a s s of lo c a l co e fficie n t sy stem s on some on T ! and h i

on Tl

t ! , where L*

are equivalent if there e x is ts some k mapping to

i and j in I su ch that the restrictio n s of l ! isom orphic.

and

hi

to

are

B e c a u s e the re strictio n s of a lo ca l co e fficie n t sy stem

a sim p licial s e t

T.

is an

via weakly homotopic maps

th is definition a ls o ap plies to categ ory of sim p licial s e ts .

S. ^ T .

StM e p ro -H , where

L

on

are isom orphic,

K is the homotopy

As d iscu sse d ab ove, if !tM e p ro-(s. s e ts ^ c )

is a pro-object of pointed, con nected sim p licial s e t s , then an isomorphism c l a s s of lo ca l co e fficie n t sy stem s on 1t!S with fibres isom orphic to a given s e t

S is equivalent to an eq uivalen ce c l a s s of homomorphisms

ttjCiTf!) = {ttjCTi)S - A u t(S ). With th e se p relim inaries, we give the following corollary of P rop osition 5 .6 . C O R O L L A R Y 5 .8 .

L et X.

s c h e m e , let U .. -> X .

b e a c o n n e c te d , lo ca lly noeth eria n sim p licia l

b e a hyper co v erin g , an d let S b e a s e t .

T h en

th ere is a natural one-to-one c o rre s p o n d en c e b etw een the s e t of isom orphism

5. HOMOTOPY INVARIANTS

c l a s s e s of locally con sta n t s h e a v e s on E t(X .)

49

with sta lk s isom orphic to

S w hose restrictio n s to Uq q are constant an d the s e t of isom orphism c la s s e s of lo ca l c o e ffic ie n t sy s tem s on ^ (A U ..)

with fib re s isom orphic to

S . In particular, an isom orphism c la s s of lo ca lly con sta n t s h e a v e s on E t(X .)

is in natural one-to-one c o rre s p o n d en ce with an isom orphism c la s s

of loca l c o e ffic ie n t sy s te m s on ( X .) e j- w hich is in natural one-to-one c o rre s p o n d e n c e with an e q u iv a le n c e c la s s of homom orphisms 771((X . ,x ) e f-) -» Aut(S) for any geo m etric point x

of X Q,

w here S is isom orphic to

the s ta lk s . Proof.

A s seen ab ove, a lo ca l co e fficie n t sy stem

sta lk s isom orphic to

By P rop osition 5 .6 , this is naturally equivalent

to a principal Aut(S)-fibration over X .

w hose re strictio n to Uq q is

Such a principal Aut(S)-fibration is e a sily seen to be equivalent

to a map of sim p licial sch em es X Q x U Q0 - U 0 0 ®S. x o

on 77(AU..) with

S is naturally eq uivalent to a principal Aut(S)-

fibration over 7t(A U ..).

triv ial.

L

X .'-* X .

sa tisfy in g 5 .3 .b .) su ch that

Such a map X.'-* X .

is equivalent to the lo cally

’

co n stan t sh eaf of s e ts on E t(X .) with sta lk s isom orphic to sending

U -» X m to the s e t of maps from U to

X^

S defined by

over X m . T his

proves the first a s se rte d eq uivalen ce of c a te g o rie s ; the secon d follow s from th is. ■ U sing C orollary 5 .8 , we now identify the cohomology of ( X .) e |- with co e fficie n ts in an abelian lo ca l co e fficie n t sy stem . P R O P O S I T I O N 5 .9 .

L et X.

b e a c o n n e c te d , lo ca lly noeth eria n sim p licia l

s c h e m e , let M b e a lo ca lly co nstant a b elia n s h e a f on E t(X .) , a nd let M a ls o d en o te the co rresp o n d in g a b elia n lo ca l c o e ffic ie n t sy stem on ( X .) e {_. T h en there is a natural isom orphism H *(X . ,M) ~ P roof.

If L

H * ((X .)e t ,M) .

is an ab elian lo ca l co e fficie n t sy stem on a sim p licial s e t S. ,

then H*(S. ,L )

is defined to be the cohomology of the com plex

50

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

in h

II s

n

fS

L (s ) ! .

If U .. -» X .

is a hypercovering of X .

su ch that the

n

lo cally co n stan t abelian shea^f M is co n stan t when re stricte d to U Q Q, then the cohomology of the bi-com plex M (U ..) is naturally isom orphic to H *( 7t(A U ..),M ). T h u s, the proposition follow s from Theorem 3 .8 by taking colim its indexed by the left cofinal su bcategory of H R R (X .) co n sistin g of U .. -> X .

su ch that M re stricte d to U Q Q is co n sta n t. ■

We ob serve that the hypothesis th at X .

be con n ected in P roposition

5 .9 is not n e ce ssa ry b e ca u se we can u se P rop osition 5 .2 to equate H *(X . ,M) with

n ae7T0 ( X . )

n a e n Q( X . )

H * ((X ? )e t ,M ).

H *(X a . ,M) and H * ((X .)e t ,M) with

6.

WEAK E Q U IV A LEN C ES, COM PLETIONS, AND HOMOTOPY LIMITS

In th is ch apter, we review the homotopy theory which has been employed in various ap p licatio n s of the e ta le to p o lo g ical typ e.

The role

of the co n stru ctio n s we con sid er is to enable us to obtain homotopy th e o retic information from the invariants of ( X .) e ^ con sid ered in C hapter 5. A fter review ing the definitions of various homotopy c a te g o rie s , we p resent the theorem of M. Artin and B . Mazur which provides n e ce ssa ry and su fficien t con dition s for a map to be a weak eq u ivalen ce in the prohomotopy categ ory [8].

We then proceed to re ca ll the Artin-Mazur pro-L

com pletion functor which en ables us to exclu d e from con sid eration homotopy information at sp ecified prim es’. Follow ing D. Sullivan, we next con sid er the Sullivan homotopy limit holim Su( ) which provides a c a t e ­ g o rical in verse limit for ce rta in p ro-ob jects in the homotopy categ o ry [69]. F in a lly , we con sid er the co n stru ctio n s A. K. B ousfield and D.M . Kan [13].

(Z/Q ^C

) and hoHm( ) of

T h e se ^-completion and homotopy

limit functors have the sig n ifica n t advantage of being “ rig id ” in the se n se that they take valu es in (s . s e t s * )

rather than in the homotopy ca te g o ry .

As we s e e , th e se co n stru ctio n s often provide a rigid version of the homotopy th eo retic co n stru ctio n s of Artin, Mazur, and Sullivan. We re ca ll that a map f : S. -» T .

in (s . s e ts ) is said to be a w eak

e q u iv a le n c e if the geom etric realizatio n of f is a homotopy eq u ivalen ce. We let

H,

the homotopy ca tego ry , denote the categ o ry obtained from

(s . s e ts ) by formally inverting the weak e q u iv a le n ce s. Sim ilarly, obtained from (s . s e t s * ) eq u iv alen ces.

We let

H*

by inverting pointed maps which are weak

(s. s e t s * c ) denote the ca te g o ry of pointed,

51

is

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

52

con n ected sim p licial s e ts and su b category of K * ).

K * c denote its homotopy categ o ry (a full

T hu s, a map in (s . s e t s * c ) is invertible in K *c

if and only if it induces an isomorphism on homotopy groups.

Although we

sh all not require their co n stru ctio n , we a ls o mention the co n stru ctio n of H o(pro-(s. s e t s * ) ) ,

a homotopy categ o ry of p ro-sim p licial s e t s , by

D .A . Edw ards and H.M . H astin gs [28]. We re ca ll that the n-th co sk e le to n functor, c o s k n : ( s . s e t s * ) -> (s . s e t s * ) ,

is the right adjoint of sk n( ) , 1 .3 ).

Unlike

sk n( ) ,

the n-sk eleton functor (a s in Definition

c o s k n( ) induces a functor c o s k n : K * ^ K* ,

b e ca u se for any pointed, con n ected sim p licial s e t T . T.

co sk n(T .) induces isom orphisms

and the homotopy groups

the natural map

^ ( T . ) -» 77'^ c o s ^n

^or k < n

77^(coskn T .) are zero for k > n . T his o b serv a­

tion permits us to give the following definition. D E F I N I T I O N 6 .1 .

to

The functor

# : ( s . s e t s * ) ^ (p ro -s . s e t s * )

sending T .

# (T .) = {c o s k n T . ; n > 0 ! exten d s to a functor # : pro-K*

by sending i s i j i e l !

pro-K*

to ic o s k n S1. ; (n ,i) e N x I } .

A map f :{ S * !- > { T ? ! in

pro-H* is said to be a w eak e q u iv a le n c e in pro-K*

if # (f ): #isM -> #\t I !

is an isomorphism in p ro -K *. ■ The introduction of weak eq u iv alen ces in pro-K* following theorem . (s . s e t s * c ) ,

is ju stified by the

As in Chapter 5 when we con sid ered o b je cts in pro-

for any iS1! e pro-K *c

we define

^ ( { s i i ) = { ^ ( S i ) ! and

H*(isi!,M) = colim H *(S! ,M) for any ab elian lo ca l co e fficie n t sy stem M on iS.1! (given by an eq u iv alen ce c l a s s of homomorphisms Aut(MQ) ,

77^(1 SM) -»

where MQ is a sta lk of M; the colim it is indexed by the left

final full su b category of I co n sistin g of th o se Aut(MQ) facto rs through

^ (i s M ) -> tt^S^)) .

j for which

^ ( i s l ! ) -»

6. WEAK EQ U IV A LE N C E S, COM PLETION S, AND HOMOTOPY LIM ITS

For any map

T H E O R E M 6 .2 (M. Artin and B . Mazur [8], 4 .3 and 4 .4 ).

f : {S.M -» |T^! in pro-H^ a .)

f

,

53

the follow ing a re e q u iv a len t.

rs a w eak e q u iv a le n c e in pro-K^ .

b .)

: 7t^({sM) -* 77^({T^!)

c .)

: 77i d s ! I) -> 77^

lo ca l c o e ffic ie n t sy stem

is an isom orphism for a ll k > 0 . is an isom orphism ; and for every a b elia n

M on{T ^ },

f* : H*({T^ S,M) -> H*(lSM,M)

is an

isom orphism . ■ We observed in C hapter 5 that the fundamental pro-group and the cohomology groups with ab elian lo ca l co e fficie n ts of (X . ,x ) e £ can be identified “ a lg e b ra ic a lly .”

Theorem 6 .2 im plies that th e se alg eb raic

invariants determ ine (X . ,x ) e ^ up to weak eq u ivalen ce in pro-K^ . As a first ap plication of Theorem 6 .2 in conjunction with Chapter 5 , we provide the following co ro llary . L et X. ,x

C O R O L L A R Y 6 .3 .

b e a p o in ted , c o n n e c te d , lo ca lly n o etherian

sim p licia l s c h e m e and let H R (X. ,x ) (X . ,x )ht H R (X. ,x )

,x)

-»

.

d eterm in es a w eak e q u iv a len c e

in p ro-X * , (X . ,x ) ht -> (X . ,x )e t . P roof. final.

We verify that the forgetful functor Let

U .. ,u

H R (X. ,x ) -> H R (X .) is left

be a pointed hypercovering of X . , x and let f ,g :U ..- > V ..

be two maps of hypercoverings of X . . Then extend to k : Spec 0 ® A [1 ] -» X Q b ecau se

f*(u ), g * ( u ) : Spec

-> VQ Q

VQ over the co n stan t homotopy Spec f}® A [l]

VQ x Spec 1) is co n tra ctib le (s e e the proof of Prop osition ' xo

3 .6 ).

Thus

h :H .. -» U ..

(k,u) : Spec H -» H .. of f,g

By C orollary 5 .7 ,

is a pointing of the left equalizer

with h^(k,u) = u . j : (X .

-»(X . ,x ) et

isomorphism on fundamental pro-groups.

in pro-H^c

induces an

By Theorem 6 .2 , it su ffice s to

E T A L E HOMOTOPY O F SIM PLIC IA L SCHEMES

54

verify that j induces an isomorphism j * :H * ( ( X . ,x ) e t ,M) ->H*((X. ,x )^ ,M ) for any ab elian lo c a l co e fficie n t sy stem

M on (X . ,x )e j. . Using the

cofin ality of HR(X. ,x ) -> H R (X .) , we identify colim

j*

H *( tt(A U ..),M ) -> colim

H RR(X.)

with the map

H *( tt(AV. .),M )

HR(X.)

which w as proved to be an isomorphism in C orollary 4 .6 . ■ If G is a group and

L

a s e t of prim es, the pro-L com pletion of G

is the system of fin ite, L -to rsio n quotient groups ( i .e ., only primes in L divide their orders) of G . By ch oosin g a sm all left final su bcategory

I

of the categ o ry of a ll su ch quotient homomorphisms, we obtain a pro-group (G )L : I -» (fin ite L -g ro u p s).

A s is ea sy to ch e ck , th is con stru ction

exten d s to a functor ( ) L : pro-(groups) -» pro-(finite L -groups) . The following theorem of M. Artin and B . Mazur provides the analagous con stru ction for p ro -K ^ . T H E O R E M 6 .4 (M. Artin and B . Mazur [8], 3 .4 and 4 .3 ).

of prim es.

T h en the in clu sio n p ro -L K s|eC -* pro-H

L et L

be a set

has a left adjoint

( ) L : p ro -K ^ -» p ro-L H *c

w here L H^c

is the fu ll su b ca teg o ry of

co n s is tin g of sim p licia l

s e t s w hose homotopy gro u p s a re fin ite L -groups.

T h e map on fundam ental

pro-groups in d u ced by the ca n o n ica l map JsM -> (isM )L com pletion map for 77^ ({SM)

is the pro-L

for any {SM e p ro -H ^ . M oreover, if M is

any a b elian lo ca l c o e ffic ie n t sy stem on ( { s ! i ) L

w hose fib re s a re fin ite

L -groups, then the ca n o n ica l map in d u ces an isom orphism H *(({sM )L ,M) h

* ({ s M,m) . ■

As a corollary of Theorem s 6 .2 and 6 .4 , we conclude the following.

6. WEAK E Q U IV A LE N C E S, CO M PLETIO N S, AND HOMOTOPY LIMITS

C o r o l l a r y 6 .5 .

L e t f : i s ! ! -» !T^ \ b e a map in pro-K.

be a s e t of p rim es.

T h en (f ) L

55

an d let L

is a w eak e q u iv a le n c e in pro-K*

if and

only if a .) j(f*)L : (771({SM ))L -* (77'1(iT^ !))L

is an isom orphism ; and

b .) for ev ery a b elia n lo ca l c o e ffic ie n t sy stem M on {T ? !

w hose

fib re s a re fin ite L -groups an d w hich is r e p r e s e n te d by a map 77’1({T ^ }) -> Aut(M0) facto ring through H*(iSM,f*M) Proof.

T^l)

}))L , f* :

The validity of a .) and b .) whenever (f ) L

in pro-K * is immediate from Theorem 6 .4 . co efficien t system

is a weak eq u ivalen ce

C o n v ersely , any abelian lo ca l

M on {T^i has a su b-system

ML whose fibres are

the maximal L -torsio n subgroups of the fibres of M. k>0,

Hk(({T ? l)L ,ML )

B ecau se

},M) ->

is an isom orphism .

Hk( ( H i ) L ,M) b e ca u s e

ML on ( { t ] S ) L

Furtherm ore, for any

({T ? I)l

is in pro-L K *c .

is a colim it of lo ca l co e fficie n t sy stem s with

finite L-groups as fibres and b e ca u se

H *((!T ^ i)L , ) commutes with

directed co lim its, we conclude using Theorem s 6 .2 and 6 .4 that a .) and b .) imply that ( f ) L

is a weak eq u ivalen ce in p ro-K ^ . ■

We now con sid er the Sullivan homotopy limit functor,

hoHmSu( ) ,

which en ables us to a s s o c ia te a sin g le homotopy type to ( ( X .) e t) L in Pr° - L K* c ‘ T H E O R E M 6 .6 (D. Sullivan [6 9 ], 3 .1 ).

prim es.

L et

P

d en o te the s e t of a ll

T h ere e x is t s a functor holimSu( ) : pro-P K *c -> K *c

together with a natural transformation holimSu( ) -> id : pro-P K *c n>pro-K*c ch a ra cteriz ed by the property that this natural transformation in d u ces a b ijectio n Homj/ (S. , holimSu( ! ' r i !)) - Horn *

j( (S. ,{T ? I ) = lira Homr, (S. , T?) * 5k

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

56

for ev ery S. holim Su(ST^ i)

in H^c

and { t ! ; j e ] \ e p r o - P H ^ .

We c h o o s e

to be a Kan co m p lex for ea ch iT^ \ e p r o - P K ^ .

for any s e t of prim es L

M oreover,

the Sullivan pro-L com pletion functor

holimSu( ) ° ( h as the property that if T .

) ^ : p ro -H ^ ->

is a pointed, c o n n e c te d sirn p licia l s e t with

fundam ental group so lv a b le of fin ite type and fin ite ly g e n e ra te d h ig h er homotopy gro u p s, then the ca n o n ica l map T . -> holim Su( ( T . ) L )

in d u c e s an

isom orphism in cohom ology H*(hoHmSu( ( T . ) L ),A ) for any fin ite a b elia n L -group A

and isom orphism s

77’*(holimSu( ( T .) L )) ^ 1

H *(T . ,A )

lim ((77.(T.)L )

0 . ■

1

x ) e j-L )*

^ (h o li m ^ C X . ,x ) e t L )) =

Unfortunately, Theorem 6 .6 does not in general enable

us to determ ine either the higher homotopy groups or cohomology groups of holim Su((X . ,x ) e t L ) .

N e v e rth e le ss, the following immediate co ro llary of

Corollary 6 .5 and the isomorphism

tt. (holim Su( *

T . H *(S. , Z /£)

id -» (Z /£ )

induces an isomorphism f* : H *(T . ,Z /£ ) —*

if and only if (Z i/t)oQ( i )

is a homotopy eq u ivalen ce.

over, ( Z / Q ^ f ) is a Kan fibration whenever f : S. -» T . b e ca u se

(Z /E )^ * ) = * ,

( Z / Q ^ S .) determ ines

where

*

More­

is s u rje ctiv e ;

is the triv ial pointed sim p licial s e t,

is a Kan com plex for any sim p licial s e t (Z / t ) J i

( ) such

S.

and

(Z / f ) J i

)

) :'(s - s e t s *) ^ ( s - s e t s *) ([1 3 ], 1 .4 .2 ).

The B ousfield-K an homotopy (in v erse) limit functor holim( ) : ( s . s e t s 1) is a functor on the ca te g o ry , s e ts for any sm all categ o ry naturality with re sp e ct to

(s . s e t s 1) ,

of “ I-diagram s” of sim p licial

I ( i .e ., functors from I to (s . s e t s ) ) . I en ab les one to extend

functorial with re sp e ct to s tr ic t maps: F : J -> (s . s e ts )

(s . s e ts )

The

holim( ) to be

if a : I -> J , G : I -> (s . s e ts ) , and

are fu n ctors, then a natural transform ation F °a -> G

determ ines a map holim (F) -> h olim (G ). A natural transform ation

F -> G

of I-diagrams with the property that F ( i ) -> G (i) is a homotopy eq uivalen ce for each

i e I determ ines a homotopy eq uivalen ce

provided that each a : I -> J

F (i)

holim (F) -> holim(G)

and G(i) are Kan co m p lexes.

M oreover, if

is a left final functor betw een sm all left filtering c a te g o rie s and

G : J -> (s . s e t s )

is such th at G(j) is a Kan com plex for each

holim(G) ->h olim (G °a)

j e J , then

is a homotopy eq u iv alen ce ([1 3 ], X I .9 .2 ).

homotopy limit of Kan com p lexes is again a Kan com plex.

The

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

58

In the following theorem , we summarize se v e ra l more useful properties ) and holim( ) . We re ca ll that a nilpotent sim p licia l s e t is

of ( Z /T ) ^

a pointed, con nected sim p licial s e t with nilpotent fundamental group which a c ts nilpotently on the higher homotopy groups. T h e o r e m 6 . 8 (A .K . B ousfield and D.M. Kan [13], 1 .7 .2 , V I .6 .2 , V II.5 . 2 , X I .7 .1 ).

For any pair of s im p licia l s e t s

S.

and T.

and any prime i ,

the ca n o n ica l map ( Z / Q ^ S . x T .) -> ( Z / Q ^ S .) x ( Z / Q ^ T . )

is a homotopy

eq u iv a le n c e with a natural right in v e rs e . If S. e (s . s e ts ^ c ) then S. -» ( Z / Q ^ S . )

e ith er is nilpotent or has fin ite fundam ental group, in d u c e s an isom orphism H ^ Z / ^ ^ S . ) , Z /£) ->

H*(S. , Z / £ ) . For any I-diagram F : I -> (s . s e t s %) F (i)

is a Kan co m p lex for ea ch

i e I,

with the property that

there e x is t s a s p e c tra l s e q u e n c e

of coho m ological type = lims |7rt (F i)| = > 77t_ s (hoH m (F)) ( w here lim °( )

is the in v e rs e limit functor,

0 < s < t

lim ^ ) :(g rp s*) -> grps)

first d eriv ed functor, a n d lims ( ) : (ab. g rp s1) -> (ab . grps) •

•

•

•

its s-th d eriv ed

«

functor) w hich c o n v e rg e s in p o sitiv e d e g r e e s p rovided that lim for a ll s ,t

with 0 < s < t . B

its

1

n

4-

E ' =0

r

The following proposition will usually be applied in the s p e c ia l c a s e that each component of each which c a s e

S? and T^ has finite homotopy groups (in

lim^ 77-m(Sa SM) = 0 for all k , m > 1 ).

erality of d iscon n ected

s ! and

We require the gen­

T^ for our d iscu ssio n of function com ­

p lexes in Chapter 11. P R O P O S I T I O N 6 .9 .

p ro-(s. se ts ^ ) S] , T^

L e t f : \Sl ; i ell -> {T I ; j 0 ,

^ holim { a T;! i .

homotopy e q u iv a le n c e ; if, in addition, c ien tly large k , P roof.

then holim(a f ) is a 1) = ^ ^or a ^

lim^

then holim |a SM and holim { a T^l a re c o n n e cte d .

We employ the co n stru ctio n s of [13].

Yn = T o tn( 11*1 T^ i) , so that

Let

X n = T o tn( I I * {s i l ) and

holim Isi i = lim SXn ! and holim IT^ I = lim {Yn S.

The s tric t map f induces a map from the “ first derived homotopy se q u e n c e s ” of [13], I X .4 .1 for iX n S

-

- "

2X ^

l

-

1Lmm W

* X n*> - " l X m } - - l X ^ l

-

m>° to th o se for {Yn i , where

n[ X^1) = image O i(X m+1)

[13], X I .7.1 and the isomorphism

lims n J l s i 1)

conclude that f induces an isomorphism and each

i > 0.

77j(Xn)

^ ( X ^ ) ) . Using

lims ^ ( { t ! !) , we ^iO ^) for e a ch

n

C onsequently, the Milnor e x a c t seq u en ce implies that

f induces isomorphisms

77-(lim X n)

77*(lim Yn) for i > 0 ,

s o that

holim(f ) re s tric ts to an eq u iv alen ce on con nected com ponents. The fa ct that

holim (f) = II holim(a f ) a or

is immediate from the o b serva-

tion that any map A ->II*lsiS fa cto rs through II* {a siS -^ II*{sil for some a.

F ix some clett, and define

'S ? = h olim \a sY\ and i/I

Then the natural maps

' T l = holim Sa T ? i . j/J

a S? -> ' s\ and aT\ -> ' t I are homotopy eq uiva­

le n ce s ([1 3 ], X I .4 .1 ), so that

holim(a f )

is a homotopy eq uivalen ce if and

only if holim( ' f ) : holim \ 'SM ^ holim i 'T ? l is a homotopy eq u ivalen ce.

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

60

Observe that a ch o ice of b a se point of holim !a SH provides {

' f : { ' S 1.}

with the stru ctu re of a s tr ic t map in p ro-(s. s e ts ^ c ) which is an

isomorphism in pro-K^ . C onsequently,

holim( ' f ) : holim !

' s\\

-> holim ! 'T ? ! re s tric ts to a

homotopy eq u ivalen ce on con nected com ponents a s s o c ia te d to any b ase point of h o lim !a S* I.

If limm 7rm( ! a S}i) = 0 for a ll m > 0 , then we co n ­

clude using the above first derived homotopy seq u en ces that { * } = tTq 'Y ^ '* for a ll

m > 0.

; hence

=

T his im plies that h olim (f) induces a

b ijectio n on con n ected com ponents, with 7r0(holim ! ' ! )

tt^

h o lim ('f )

770(holim { 'SM) =* lim 1 7Ti ' X m e*

is a homotopy eq u ivalen ce.

If

lim^- 1 77-^({a SM) = 0 for k su fficien tly larg e, then th ese first derived homotopy seq u en ces imply th at \tt^( X n)S is M ittag-L effler so that holim !'S.M 0 , i h o lim (Z /£)n(T .) eq uivalen ce for any

T. e

. ■

is a homotopy

7.

FIN IT EN ESS AND HOMOLOGY

In this ch apter we co n sid er noeth eria n sim p licia l s c h e m e s ( i .e ., sim p licial sch em es which are noetherian in e a ch dim ension).

As we

verify in C orollary 7 . 2 , the e ta le to p o lo g ical type of su ch a noetherian sim p licial schem e is weakly equivalent to a p ro-object in the homotopy categ ory of sim p licial s e ts which are finite in e a c h dimension.

In

Theorem 7 . 3 , we g en eralize to sim p licial sch em es the criterion of M. Artin and B . Mazur th at the homotopy pro-groups of the e ta le top ologi­ c a l type be pro-finite.

In Prop osition 7 . 5 , we show that pro-ob jects of

finite ab elian groups are anti-equ ivalent to torsion ab elian groups.

Under

th is an ti-eq u iv alen ce, the homology pro-groups with c o e fficie n ts in the dual of a sh eaf of co n stru ctib le ab elian groups (a s defined in Definition 7.4)

correspond to cohom ology.

A theorem of P . D eligne provides

exam p les of sim p licial sch em es with finite homology groups with various c o e fficie n ts. P ROPOSITION 7.1.

sch em e.

L et X. , x

b e a p o in ted noeth eria n sim p licia l

T h en the fu ll s u b c a te g o rie s n H R (X .) C H R (X .) , nHR(X. ,x ) C H R(X. ,x )

c o n sistin g of hyper co v e rin g s U .. ^ X . s ,t > 0

with Ug ^ noeth eria n for a ll

(“ noetheria n hy p er c o v e rin g s ” ) a re left filterin g and the in clu sio n

functors a re left final. P roof.

If U -> X n and V -> X n are e ta le with U and V noetherian,

then th ese maps are of finite typ e; thus, the fibre product of U -» X n X n in E t ( X .) ,

U is noetherian if and only if U is a

union of finitely many con n ected com ponents. prove for any pointed hypercovering V .. ,v -» U .. ,u su ch th at Vg t

T h erefore, it su ffice s to

U .. ,u -> X .. ,x

Us t

the e x is te n c e of

is the inclusion of finitely many

com ponents of Ug ^ for all s ,t > 0 and V .. ,v -> X .. ,x hypercovering.

We define

VQ Q

is a pointed

U Q Q to be the pointed inclusion of

finitely many com ponents of U Q Q which co v er X Q. P roceed in g inductively, we define

Vg t -> Ug t to be the in clu sion of those co n ­

n ected components of Ug t which are in the image of Vg l t or Vg t l under some d egeneracy map of U ..

together with finitely many co n ­

n ected com ponents of the fibre product of (cosk^ ^Vg 0

= X g = c o s k jV g ) . ■

As an immediate corollary of Prop osition 7.1 and C orollary 6 .3 , we obtain the following fin iten ess property of the weak homotopy type of (X . ,x )e t . C O R O L L A R Y 7 .2 .

L et X. ,x

sim p licia l s ch em e .

D e fin e (X .

homotopy type of X . , x ,

b e a pointed, c o n n e c te d , n o etherian in pro-H^ ,

the noeth eria n (e ta le )

to be

(X . ,x ) nfa = 77°A : nHR(X. ,x ) -> (s . s e ts ^ ) . T h en (X . ,x ) n^. is a p ro -ob ject in the homotopy ca tego ry of po in ted sim p licia l s e t s w hich a re fin ite in ea ch d im en sio n .

Furtherm ore, the

natural maps (X- >x )nht

(X - »x )ht

^X - ’x ^et

a re w eak e q u iv a le n c e s in pro-K^ . ■ We remind the reader that a schem e

X

is said to be geom etrically

unibranched if the in tegral clo su re of e ach of the lo ca l rings of X

(sta lk s

7. FIN IT E N E SS AND HOMOLOGY

65

of the stru ctu re sh e a f in the Z a risk i topology) is a ls o a lo c a l ring.

In

p articu lar, if e ach of th e se lo c a l rings is already integrally clo se d (as is the c a s e if they are regular lo ca l rin g s), then X

is g eom etrically

unibranched. The following theorem is an e a sy gen eralizatio n of a theorem of M. Artin and B . Mazur ([8 ], 1 1 .2 ).

We sh all find th is theorem p articularly

useful when we co n sid er function co m p lexes. T H E O R E M 7 .3 .

L et X. ,x

be a pointed, n o eth eria n sim p licia l s c h e m e

su ch that X n is c o n n e c te d and g eo m etrica lly u n ib ra n ch ed for ea ch T h en any pointed, noeth eria n h y p erco v erin g property that

A U . . ) , u)

U .. , u

is fin ite for e a ch

k > 0.

X. ,x

n > 0.

ha s the

C o n seq u en tly ,

77k ((X . ,x ) e ^.) is a pro-finite group ( i .e ., isom orphic in pro -(grp s) to a pro­

o b ject in the ca teg o ry of fin ite g ro u p s). Proof. any

F o r any pointed, noetherian hypercovering U .. , u -» X . , x and

n > 0 , Un -» X n is a pointed, noetherian hypercovering so that the

Artin-Mazur theorem im plies that (B e c a u s e U -> X

^ ( ^ ( b ^ )) is finite for e a ch

k > 0.

X n is geom etrically unibranched, any con n ected e ta le open

is irreducible.

th at th is implies that

The startin g point of their proof is the observation 7r(Un ) =

x Spec K ) , where

K is the field

' xn of fractio n s at the generic point of X n .) B e c a u s e e a ch

?KUn ) is co n ­

nected by P rop osition 5 .2 , we may apply the homotopy s p e ctra l seq uen ce of [12], B .5 , to con clu d e th at the homotopy groups of a ls o finite.

77-^(t7( A U . . ) , u )

are

C orollary 7 .2 now im mediately im plies th at ^ ( ( X . ,x ) j.) is

pro-finite for each

k > 0. ■

The n e c e s s ity of the hypothesis that ea ch

X n be con nected in

Theorem 7 .3 can be readily understood by examining the c a s e in which X. = S p ecK ® S 1 . Our approach to the homology of an ab elian co p resh eaf is motivated by C orollary 3 .1 0 .

We sh a ll find it su fficien t to co n sid er co p re sh e a v e s

66

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

which commute with finite d isjoint unions b e ca u s e we sh a ll re s tric t our attention to noetherian sim p licial sch e m e s; the reader should observe that the dual of a sh eaf does not commute with arbitrary d isjoint unions. D E F I N I T I O N 7 .4 .

Let X.

c o p re s h e a f on E t ( X .) property that F o r any

be a sim p licial sch em e.

is a functor

P : E t ( X .)

An a b elia n

(ab. grps) with the

P (U H V ) - P ( U ) e P ( V ) for any U ^ X n , V -+Xn in E t ( X .) .

i > 0,

the i-th hom ology pro-group of X .

abelian cop resh eaf

P

with valu es in the

is the pro-group

H ^ X . #P ) = Hi o P ( ) : H R (X .) -> (ab. grps) sending

U .. -» X .

to the i-th homology group of the bicom plex

P ( U ..) . ■

We sh all be e sp e c ia lly in terested in the following two exam ples of abelian co p re sh e a v e s. E t ( X .) ,

If M is a lo ca lly co n stan t ab elian sh eaf on

we define M ° :E t ( X .) -» (ab. grps)

by settin g M °(U) = ®M°(Ua ) , con n ected com ponents M restricted to

Ua

Ua

where the d irect sum is indexed by the

of U and where M °(Ua ) = M(Ua ) whenever

is co n sta n t and M °(Ua) = 0 otherw ise; we define

M °(U) ->M°(V) a s s o c ia te d to a map U -> V in E t ( X .) to be the map whose restrictio n to M °(Ua ) is given by the in verse of M(V^g) -> M(Ua ) , where Vjg is the component of V containing the image of Ua restricted to Ua If F F (V ),

if M

is co n stan t.

is an abelian presheaf on E t(X .)

such that F (U II V) = F (U ) x

then we define the dual c o p re s h e a f F v : E t ( X .) -» (ab. grps)

by F V(U) = H om (F(U ), Q /Z ) . If P : E t ( X .) -> (ab. grp s) tak es v alu es in the categ o ry of finite abelian groups (denoted -> H R (X .) im plies th at

(f. ab. grps) ), then the left finality of n H R (X .) H^(X. ,P )

is pro-finite for e ach

i > 0.

Such

67

7. FIN IT E N E SS AND HOMOLOGY

pro-finite abelian groups are not so unfam iliar, as we show in the next proposition. P

r o po sitio n

7 .5 .

L e t ( ) v = Hom(

,Q /Z ) .

T h en the functor

colim o ( ) v :(p ro -(f. ab. g rp s ))0 -» (tor. ab. grps) is an e q u iv a len ce of c a te g o rie s from the o p p o site ca tego ry of the ca tego ry of p ro -ob jects of fin ite a b elia n groups ( “ pro-finite a b elia n g ro u p s ” ) to the ca tego ry of torsion a b elia n gro u p s.

In pa rticu la r, {A- \ e pro-(f. ab. grp s)

is fin ite ( i .e ., isom orphic to a fin ite group) if a n d only if colim {A^i fin ite if and only if lim {A -l P roof.

is

is fin ite.

B e c a u s e any torsion abelian group A is the colim it of its finite

subgroups, b ecau se the categ o ry of finite subgroups of A is left d irected (where B

maps to C in this category if and only if B

and b ecau se

co n tain s

C ),

( ) v is a (co n trav arian t) involution of (f. ab. grp s), the

functor colim ° ( ) v is e s se n tia lly s u rje ctiv e . i elS -» !H j; j e ] I

T o prove colim ° ( ) v is faithful, co n sid er f,g : in pro-(f. ab. grps). for e a ch

j £J

Then f = g if and only if fj = gj

if and only if f J = g j

if and only if f^ = g j

in colim Hom(G-,Hj)

in colim Hom(H^,G^) for each

in Hom(Hj/, c o lim \G^\) for e a ch

j e]

j £J

(b e ca u se

Hj' is fin ite) if and only if colim fj' = colim g^ in Hom(colim {H ^l,colim {G^i). To prove c o l i m o ( ) v is fully faithful, co n sid er f : colim colim {G^i and w rite

1

f = lim f - with f •: colim lG-v ! . Since

fin ite,

fj facto rs through

G^

fin ite, th is image facto rs through some

is

the image

fa cto rs through some map f • ^

J>K

map determined by f ^ u , J fK

J

1

Gj£ -> colim iG^S. J

is

1

T hu s, J

be the

then the fa c t that colim ° ( ) v is faithful im-

plies that ifj^l: iGjJ -» {H jl is a w ell-defined map and that colim o ( ) v({fj/ !) = f .

J

of some G ^ -> colim {G ^ l;

G^ . If we le t f^ : lG -! -> HK

-»

fj

and sin ce

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

68

In p articu lar, the fa c t th at colim ° ( ) v is an eq u ivalen ce im plies th at colim {A^S = A is finite if and only if |A^| is isom orphic to a finite group A .

Since

(colim {A ^ i^ = lim { A j l , colim {A^S is finite if and only

if lim{A*S is finite. ■ We next u tilize the eq u ivalen ce of P rop osition 7 .5 to re la te homology to cohomology. schem e

We remind the reader that a sh eaf F

on a noetherian

(with the e ta le topology) is said to be c o n stru ctib le if there

X

e x is ts a finite c o lle c tio n of lo ca lly clo se d su bschem es disjoint union is

X

such that F

co n stan t with finite s ta lk s . c ia l schem e and F

re stricte d to e a ch

More gen erally, if X .

is a sh eaf on E t ( X .) ,

X-

then F

F (U ) is a finite s e t for any

whose

is lo cally

is a noetherian sim pli­ is said to be co n ­

stru ctib le if F n is co n stru ctib le on E t(X n) for e a ch co n stru ctib le, then

iX-S of X

n>0.

If F

is

U -* X n in E t ( X .) with

U noetherian. P R O P O S I T I O N 7 .6 .

F

L et X.

b e a noeth eria n sim p licia l s c h e m e , and let

b e a co n stru ctib le a b elia n s h e a f on E t ( X .) .

For ea ch i > 0 ,

th ere is

a natural duality isom orphism (colim o (

) v) (H -(X . , F v))

-

Hi(X . , F ) ,

so that H -(X. , F V) is fin ite if a n d only if H*(X. , F )

is fin ite.

M oreover,

if M is a locally co n sta n t c o n stru ctib le a b elia n s h e a f on E t ( X .) , for a ll i > 0

th ere a re natural isom orphism s (colim ° ( ) v)(H i(X . ,M0))

P roof. If U .. -> X . sh e a f, then

then

*

Hi (X . ,M) .

is a noetherian hypercovering and F

a co n stru ctib le

( F V( U ..) ) V is isom orphic to F ( U ..) s o that (H -(F V( U ..) ) ) V ^

H * (F (U ..)) for any i > 0 .

C onsequently,

colim (H i( F v(U ..) ) v) s* colim H ^ U . . ) ) where the co lim its are indexed by n H R (X .).

~

H ^X. ,F )

7. FIN IT E N E SS AND HOMOLOGY

69

If M is a lo ca lly co n sta n t, co n stru ctib le abelian sh eaf on E t ( X .) , then M °(U ..) is naturally isom orphic to MV(U ..) whenever U .. noetherian hypercovering su ch that M re stricte d to

is a

U Q 0 is co n sta n t.

C onsequently, colim (H i (M °(U --))V) “

colim (H i(Mv( U ..) ) v)

“

H ^ X . ,M) . ■

Using P rop osition 7 .6 , we immediately conclude the homology analogue of the sp e c tra l seq u en ce of P rop osition 2 .4

E s ,t = Ht (X s , F v s) = > Hs + t( X , F v) w henever X .

is noetherian and F

is co n stru ctib le .

We ca n a ls o co n ­

clu d e, for exam ple, the homology analogue of P rop osition 3 .7 a sse rtin g that H#(X . , F V) whenever U .. -> X .

~ H *(A U .. , F V)

is a noetherian hypercovering and F

is co n stru ctib le .

We conclude th is ch apter with the following theorem whose proof is an immediate con seq u en ce of P . D elig n e’s criterion for the fin iten ess of cohomology ([2 4 ], “ F in itu d e ” 1 .1 ) (extended to sim p licial sch em es using P rop osition 2 .4 ) and P rop osition 7 .6 . T H E O R E M 7 .7 .

w here R

L et X.

be a sim p licia l s c h e m e of fin ite type over Spec R ,

is a co m p lete d is c r e t e valuation ring with a lg eb ra ic a lly c lo s e d

re s id u e fie ld .

T h en for a ll i > 0 , H ^ X . , F )

for any co n stru ctib le a b elia n s h e a f F

an d H ^ X . , F V) are fin ite

on E t( X .) with sta lk s of order

in v ertib le in R . ■ The condition on the orders of the sta lk s of F n e ce ssa ry .

in Theorem 7 .7 is

F o r exam ple, H ^ S p e c k[x], Z /p ) is infinite whenever k is an

infinite field of c h a ra c te r is tic

p.

8.

COMPARISON O F HOMOTOPY T Y P E S

In Chapter 5 , we identified various homotopy invariants of the e ta le top o lo gical type

(X . ,x ) e j. in terms of alg eb raic in variants.

In Chapter 6,

we d escrib ed the hom otopy-theoretic co n te x t in which th e se invariants play a cen tral role.

We now proceed to employ th is m aterial to tra n sla te

various theorem s concerning e ta le cohomology groups and fundamental groups into theorem s concerning the homotopy type of (X . ,x )e j.. We begin by comparing in P rop osition 8.1 the homotopy type of (X . ,x )e £ to that of (A U .. ,u )e |., where U .. , u -» X . , x

is a pointed hypercovering.

The proof of P rop osition 8.1 is rep resen tativ e of the method of proof of e ach of the re su lts in this ch ap ter.

F o r a sim p licia l schem e X .

over the

com plex numbers, Theorem 8 .4 p resen ts the very useful com parison of the homotopy type of (X . ,x )e t with the homotopy type of its underlying sim p licial sp a ce with the “ c l a s s i c a l to p o lo g y .”

Prop ositions 8 .6 and 8 .7

obtain homotopy th eo retic co n clu sio n s from the proper b ase change theorem and smooth b ase ch ange theorem for e ta le cohom ology.

F in a lly , in

Prop osition 8 .8 , we in v e stig a te reductive group sch em es and their c l a s s i ­ fying s p a c e s . In this first proposition, we conclude that from a homotopy-theoretic point of view a sim p licial schem e may be replaced by one of its hyper­ cov erin g s.

The u sefu ln ess of this co n clu sio n lie s in the fa ct that the

hypercovering may c o n s is t of sch em es which are sim pler (being more lo ca l) than the original sim p licial sch em e.

F o r exam ple, this is the b a sic

observation underlying the d iscu ssio n of tubular neighborhoods of Chapter 15.

A somewhat weakened form of the following proposition was proved by

D. C ox in [18], IV .2.

70

8. COMPARISON OF HOMOTOPY T Y P E S

P R O P O S I T I O N 8 .1 .

L et X. ,x

71

b e a pointed, c o n n e c te d , lo ca lly n o etherian

sim p licia l s c h e m e , and let g : U .. , u -» X . , x

be a p ointed h y p er co v erin g .

T h en ge t : (A U .. ,u)e t - ( X . ,x )et is a strict map of p ro-(s. s e ts ^ c ) w hich is a w eak e q u iv a len ce in pro-H^ . In particular, for a ll i > 0 , g^ : 7t-((AU.. ,u ) t )

^ ( ( X . ,x ) t)

is an

isom orphism , and if th e s e pro-groups a re pro-finite, then (h o lim °S in g .( )o| |) ( g ) : holim (Sing.(|(AU.. ,u )e t |))->hoUm(Sing.(|(X. ,x )e t |)) is a homotopy e q u iv a le n c e (w h ere

| |: (s . s e ts ) -» (top. s p a c e s )

is the

geo m etric rea liza tio n functor and Sing.( ) : (top. s p a c e s ) -* (s . s e ts )

is the

sin gu la r functor a s d is c u s s e d in [42]). P roof.

By P rop ositio n s 3 .7 and 5 .9 ,

H * ((X .)e j.,M)

g : A U .. -> X .

H *((A U ..)e |_,M) for any abelian lo ca l co e fficie n t system

M on ( X .) e £ . By Theorem 6 .3 , to prove that in pro-K ^,

induces an isomorphism

ge j. is a weak eq uivalen ce

it su ffice s to prove th at g a lso induces an isomorphism

771((A U .. ,u)e t ) -» ^ ( ( X . ,x )e t ) .

By Prop osition 5 .6 , it su ffice s to prove

that g induces an eq u ivalen ce of c a te g o rie s any group G .

I1(X. ,G)

II(A U .. ,G ) for

As re ca lle d in the proof of P rop osition 5 .6 , th ere is an

eq u ivalen ce of ca te g o rie s

I1(XS,G)

n (U g ,G) for any

s > 0.

C o n se­

quently, the required eq u ivalen ce follows from Lemma 5 .4 and P roposition 5 .5 , which provide the interm ediate eq u iv alen ces < r i ( X 0,G); d. data > and < II(U a ,G); d. data >

II(X . ,G) I1(AU.. ,G ) .

Theorem 6 .2 now im plies th at ge ^ induces an isomorphism on homotopy pro-groups, w hereas P roposition 6 .9 im plies that (holim °S in g. ° | |)(g) is a homotopy eq uivalen ce (b e ca u se the natural map S. -» Sing. (|S. |) is a weak eq uivalen ce with Sing. ( |S. |) a Kan com ­ plex for any sim p licial s e t S. ). ■ Most ap p licatio n s of e ta le homotopy theory have u tilized the “ C ech to p o log ical t y p e / ’ A (X. ,x ) re|., a s defined preceding D efinition 4 .4 .

We

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

72

verify th at A (X. ,x )

^ has the sam e weak homotopy type as

for “ m o st” pointed sim p licia l sch em es L et X. ,

P R O P O S I T I O N 8 .2 .

X

X. , x .

b e a pointed, c o n n e c te d , lo ca lly n o etherian

sim p licia l s c h e m e , a n d let A ° ( X . >x )ret (a s d e fin e d a b ov e).

(X . ,x )e t

d en o te the C e c h top o logica l type

T h en there is a natural stric t map of p ro-(s. s e ts ^ c ) (X . ,x )et -> A o( X . ,x )ret

w hich in d u ces an isom orphism ^ ( ( X . >x )e t) ^

77’1(A o (X . ,x )re|.). Fu rther­

more, if X n is q u a si-p ro jectiv e over a n o eth eria n ring for e a ch n > 0 , then this map is a w eak e q u iv a le n c e in pro-K^ . P ro o f. The a sse rte d s tr ic t map is induced by the rigid analogue of cosk ^ *( ) : C (X .) -> H R (X .) . The fa c t that this map induces an isom or­ phism on fundamental pro-groups follow s d irectly from C orollary 5 .7 . Arguing a s in the proof of Proposition 5 .9 , we e a s ily conclude the natural v isomorphism H *(X . ,M) H*(A o (X . ,x ) re|.,M) for any lo ca lly co n sta n t, abelian sh eaf M on E t ( X .) . noetherian ring for e a ch

T hu s, if X n is q u asi-p ro jectiv e over a

n > 0,

then Theorem 3 .9 implies that (X . ,x > e t -

A o (X . ,x )re|. induces an isomorphism in cohomology with any ab elian lo ca l c o e fficie n ts .

C onsequently, for a pointed sim p licial sch em e, the

fa ct that (X . ,x ) e t ^ A ° (X . ,x ) re£ is a weak eq u ivalen ce in pro-H^ follow s from Theorem 6 .2 . ■ We next proceed to co n sid er sim p licial sch em es C ( i .e .,

X.

of finite type over

X n is of finite type over Spec C , where C denotes the com ­

plex numbers).

If X

com plex points of X finite type over

C,

is of finite type over C , then X to P is the s e t of with the usual (a n a ly tic) topology; if X . then x ! ° P

is of

is the sim p licia l s p a c e ( i .e ., the sim pli­

c ia l object of top ological s p a c e s ) with

(x t°P )n = X * °P .

We re ca ll that the g eo m etric rea liza tio n of a sim p licial s p a ce is the quotient of the top o lo g ical sp a c e

II n> 0

T . , |T. |,

Tn x A[n] by the eq uivalen ce

73

8. COMPARISON OF HOMOTOPY T Y P E S

relation

( t ,a ( x ) ) ~ (a (t), x) for any a :A [n ]-> A [m ]

in A (where Tn x A [n ]

is given the product topology and A[n] = lx = ( x 0,-*- ,x n) : Sx^ = 1, x- > 0 j c R n+1) We a lso re ca ll th at if S..

is a b i-sim p licial s e t and if In

a s s o c ia te d sirnp licial s p a c e , then

|A(S..)|

|S

|l is the

(the geom etric re a liz a tio n of

the diagonal sirnp licial s e t) is homeomorphic to

|in h> |Sn |!| (cf. [65], 1).

We now give a te ch n ica l lemma relatin g the singular cohomology of the geom etric realizatio n of a sirnplicial sp a c e

T.

to its sh eaf cohom ology.

In analogy with D efinition 1 .4 , we define the lo ca l hom eom orphism s it e L h (T .) as follow s.

As a ca te g o ry ,

homeomorphisms W -> Tn for some

L h (T .) has o b je cts which are lo ca l n > 0;

a map in L h (T .)

is a commuta­

tive square W ---------------

Z

Tn -------------►Tm with Tn

Tm a sp ecified stru ctu re map of T . ; a covering of W -> Tn

is defined to be a family of lo ca l homeomorphisms w hose im ages of W- in W co v er W.

{Wj->Wi over Tn

As in Definition 2 .3 , for any

i > 0 we define Hj;h(T . , ) : A bSh(T .) -» Ab to be the i-th right derived functor of the functor sending an ab elian sh eaf F

on L h (T .) to the kernel of the map d ^ - d * : F (T Q) -» F ^ ) .

L E M M A 8 .3 .

L et T.

b e a sim p licia l s p a c e with

paracom pact for

ea ch n > 0 .

For any lo ca lly co n sta n t a b elia n s h e a f M on L h ( T .) ,

a re natural isom orphism s H*h(T . ,M) w here Sing. ( T .) Sing t(Ts ) .

«

H *(A (S in g .(T .)),M )

«

H*(Sing. (|T.

|),M)

is the b isim p licia l s e t g iv e n in b id e g re e s ,t

by

th ere

74

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

P roof.

B e c a u s e every lo c a l homeomorphism W -> Tn is covered by a

family

fW- -» WS with e a ch

-> W -> Tn an open immersion, the ca te g o rie s

of ab elian sh e a v e s on L h (T .) and on the analogous s ite immersions are equivalent.

We define a sh e a f

O i(T .)

of open

on (3 i(T .) for any

p > 0 as the sh eaf a s s o c ia te d to the presh eaf sending an open immersion W ^ T n to

CP(Sing. (W),M) ([1 4 ], 1.7).

Then M -> S*M is resolution with

the property that (§P M )n is a c y c lic on G i(T ) for any I I I.l).

p,n > 0 ([1 4 ],

U sing the lo ca l homeomorphism analogue of Prop osition 2 .4 , we

con clu d e that H ^ ( T . ,M) is naturally isom orphic to H * (§ M (T .)). On the other hand, the natural map C *(S in g. (Tn),M) -» S M(Tn) induces an isomorphism in cohomology ([1 4 ], 1 .7 ) for each natural map of b icom p lexes phism in cohomology

n > 0 , so th at the

C *(Sing. (T .),M ) -» S M (T.) induces an isom or­ M ) • T hus, to com plete

H*(A(Sing. (T .)),M )

the proof, it su ffice s to verify th at the natural map A(Sing. (T .)) -> Sing. (|T. |) induces an isomorphism in cohom ology. paring the cohom ology of

T his is readily shown by com ­

|A(Sing. (T.))|

tion of the sim p licial s p a c e

!n

with th at of the geom etric re a liz a ­

|Sing. (T )|! with that of

|T.| . ■

The main ingredient of the proof of the following theorem is the “ c l a s s i c a l com parison theorem ” for e ta le cohomology proved by M. Artin and A. G rothendieck and the “ Riemann e x is te n c e theorem ” proved by H. Grauert and R . Remmert ([7 ], X I .4 .3 ).

V arious (w eaker) versio n s of

Theorem 8 .4 have been proved first by M. Artin and B . Mazur ([8 ], 1 2 .9 ), then by the author, R. Hoobler and D. R e cto r, and D. C ox. T H E O R E M 8 .4 ( C o m p a r i s o n T h e o r e m ) .

L et X. , x

sim p licia l sc h e m e of fin ite type over C , a s s o c ia t e d sim p licia l s p a c e (a s a bo v e).

be a po in ted , c o n n e c te d

and let X^°P, x

d en o te the

T h e re e x is t strict maps of

pro-(s. s e t s * c )

(X . ,x ) e t « L (X . ,x ) s et - L s in g .(| x J ° P ,x | ) su ch that r

is an isom orphism in pro-H^ , and (p )P

le n c e in pro-K^ , w here P

is a w eak eq u iv a ­

is the s e t of a ll prim es ( c f . C orollary 6 .5 ).

8. COMPARISON OF HOMOTOPY T Y P E S

P roof. We define

(X . ,x )g

A °Sing. ( ) o ( sending U .. -> X .

75

in p ro-(s. s e ts ^ ) to be the functor

)toP o A : H R R ( X . ) - ( s . s e t s * )

to the diagonal of the b isim p licial s e t

Sing. ( A u !? ^ ) .

The map r is the com position of the natural map 77: A(Sing. (A u !? ^ )) -> A(Sing. (x!" 0P)) (for any

U .. -» X . ) and the ca n o n ica l homotopy eq uiva­

len ce A(Sing. (X .)) -* Sing. (|xt°P|) (d iscu sse d in the proof of Lemma 8 .3 ). Using d escen t for principal homogeneous s p a c e s over X g° P ,

the argument

of [8 ], 10, ap p lies to show that II(X g0P,G) is equivalent to II(Ug0 P ,G ). As argued in the proof of P rop osition 8 .1 , th is im plies that II(A u t?P ,G ) is equivalent to I I ( x ! 0 P ,G ), which im plies that

^ (lA u l'P ^ u l) 2*

771(A(Sing. (u t? P ,u ))) is isomorphic to 7r 1(A(Sing. (X t° P ,x ))) ^ 77-1(| x t° P ,x | ). 77 induces an isomorphism of fundamental groups.

C onsequently,

As argued in C orollary 5 .8 , an ab elian lo c a l co e fficien t system

M on

A(Sing. (X t°P )) is in one-to-one correspon d ence with a lo ca lly co n stan t ab elian sh eaf on L h ( x t ° P ) . By Lemma 8 .3 and the lo ca l homeomorphism analogue of P rop osition 3 .7 ,

77 a ls o induces an isomorphism in cohomology

with ab elian lo ca l c o e ffic ie n ts . len ce for any

U ..

T herefore,

77 is a weak homotopy equiva­

in H R (X .) , so that r is an isomorphism in

X.

Pro-ft* • The map p is induced by the natural transform ation A oSing. ( ) ° ( defined by sending a : A[k] ->

)^0 P ° A ^ 77°A (A(Sing. (A U .t0 ^)))^ to the co n ­

nected component in (tz(AU ..))jc containing the image of a . T o prove that (p )P is a weak eq u iv alen ce in p ro-K ^, we facto r p a s the com posit ion S o y o ^ - l : (X . ,x )s>et - (X . ,x ) s i h - (X . ,x ) £h -> (X . ,x ) et defined as follow s. the s ite

E t(X .)

Let

H R R (X ?°P) be defined a s in P rop osition 4 .3 with

rep laced by L h (X t°P ) . Define (X . ,x )g g^ to be

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

76

A oSing. ( )°A :H R R (X ^ ° P ) -> (s. s e ts ^ ) and define

(X . ^ x)^

77o A : HRR(X^°P) -> ( s . s e t s . Define £ : (X . ,x )s>01

to be

(X . ,x ) s>et to be

the isomorphism in pro-K^ (a s argued ab ove, both are isom orphic to Sing. (|xJ°P,x|) ) determined by the forgetful functor

H R R (X .) ^ H R R (x t°P ).

Define y e x a c tly as we defined p , and define 8 to be a ls o induced by T hen, p ° /3 = < 5 ° y :(X . ,x )s

the forgetful functor.

As in P rop osition 5 .6 , we identify s o that y

tz^

-» (X . ,x )e t .

X . j X )^ ) with ^ ( I X ^ ^ x l ) ,

induces an isomorphism on fundamental pro-groups. As in

Prop osition 5 .9 , we identify

H ^(X^°P,M ) with H *((X . ,x)g^,M) for any

lo cally co n stan t ab elian sh eaf on L h (x !:0P ) , s o th at Lemma 8.3 implies that y induces an isomorphism in cohomology with ab elian lo ca l c o e fficie n ts.

T h u s,

y is an isomorphism in pro-M^ .

F in a lly , the Riemann e x is te n c e theorem and the usual d e sce n t argument relatin g

II(X^0 P,G) to < I ^ X ^ P .G ); d. data > and II(X. ,G) to < II(X 0,G);

d. data > imply that 8

induces an isomorphism on the

of the fundamental pro-groups.

pro-P com pletions

By the c l a s s i c a l com parison theorem and

P rop osition 2 .4 , we con clu d e that 8

induces an isomorphism in cohomology

with ab elian lo ca l co e fficie n ts w hose s ta lk s are finite.

T hu s, ( ( Z /£ ) J S i n g . (|x‘ °P ,x | )).

P roof. By P rop osition 6 .1 0 and C orollary 7 .2 , it su ffice s to prove that p and t induce homotopy eq u iv alen ces holimSU((X . , x & ) - holimSU((X . , x £ e t) - hoHmSu(Sing. |X?°P,x|£) .

T his follow s immediately from Theorem 8 .4 and C orollary 6 .7 . ■

77

8. COMPARISON OF HOMOTOPY T Y P E S

We next provide the homotopy th eo retic v ersion of the proper b a se change theorem in e ta le cohomology ([7 ], X II.5 .1 ).

We re ca ll th at a s trict

h e n s e l lo ca l ring is a lo ca l ring with a sep arab ly c lo se d residue field which s a tis f ie s H e n s e l’s lemma ([5 9 ], 1.4).

Such a lo ca l ring R has the

property that (Spec R )et is co n tra ctib le . P R O P O S I T I O N

L et

8 .6 .

R

b e a stric t h e n s e l lo ca l ring, and let

a c o n n e c te d sim p licia l s c h e m e over Spec R proper for ea ch

n > 0.

re s id u e fie ld of R ,

be

X .

su ch that X n -> S p e cR

is

L e t K be a sep a ra b ly c lo s e d fie ld containing the

an d let i : Y . -> X .

b e d e fin e d by Y = X n n

For any geo m etric point y

of Y Q,

x

n

Spec K.

S p ec R

the strict map of p ro-(s. s e ts ^ c )

i ' (Y- ,y )e t - ( X - >y)et is su ch that ( i ) P

is a w eak e q u iv a le n c e in pro-K^ , w here P

of a ll p rim es. C o n s e q u e n tly , for any prime £ ,

i

is the s e t

in d u ces a homotopy

e q u iv a le n c e i:h o li m o (Z /£ )oo((Y . ,y )e t) -> holim ° ( Z / £ ) oo((X . ,y )e t ) . Proof.

The proper b ase ch ange theorem im plies that i g induces an

eq u iv alen ce of c a te g o rie s

I1(YS ,G)

II(X S,G) for any finite group G .

By Lemma 5 .4 and P rop osition 5 .6 , th is im plies that

i induces a b ijection

Hom(77'1(X . ,x ),G ) -> Hom(771(Y . ,y ),G ) for any finite group G .

Moreover,

the proper b ase change theorem and P rop osition 2 .4 imply that i induces an isomorphism H *(X . ,M)

H *(Y . ,M) for any lo ca lly co n sta n t, co n ­

s t r u c t i v e ab elian sh eaf M on E t ( X .) . Prop osition 5 .2 . imply that

In p articu lar, Y .

is con nected by

By P rop osition 5 .9 , th e se cohomology isomorphisms

i induces an isomorphism

ab elian lo c a l co efficie n t sy stem

H * ((X .)e t ,M) -» H* ( ( Y . ) e t ,M) for any

M on ( X .) e |. with finite fib res.

By

Corollary 6 .5 , th is im plies that ( i ) P is a weak eq uivalen ce in p ro-K ^ . C onsequently, P rop osition 6 .1 0 and C orollary 7 .2 imply that holim ° (Z/Q oo((Y- ,y )e t) -> holim o (Z /£ )oo((X . ,y )e t ) is a homotopy eq u iv alen ce. ■

78

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

The proof of P rop osition 8 .6 ap plies a s well to prove the following theorem provided we employ the proper, smooth b a se change theorem ([7 ], X V I.2 .2 ) rather than the proper b a se change theorem . PROPOSITION 8 .7 .

L et R

b e a s trict h e n s e l lo ca l domain, and let X .

b e a c o n n e c te d sim p licia l s c h e m e over R proper and smooth for ea ch

n > 0 . L et

the stru ctu re map X Q ^ Spec R ,

s u c h that X n -> Spec R

e : Spec R

let x : Spec

is

X Q b e a se c tio n of

-> S p e cR -> X Q b e a

geo m etric point over the c lo s e d point of S p e cR , and let z : 12z -> S p e cR -> X Q b e a geo m etric point over the g e n e r ic point of S p e cR

(s o that any

e ta le neighborhoo d U -> X Q of x

is a ls o an eta le n eigh bo rh o o d of z ).

L e t j : Z. -» X .

= X

b e d e fin e d by Z

11

sep a ra b ly c lo s e d fie ld containing R .

x S p e c F , w here F 11 Spec R T h en j

is any

in d u c e s s trict maps in

p ro-(s. s e ts ) a n d pro-K ^ j : ( Z .) et - (x -)e t - j : (z - >z )ht - (X - ’x \ t su ch that ( j ) L

is a w eak e q u iv a le n c e in pro-M^ , w here L

is the s e t of

a ll prim es e x c e p t the re s id u e ch a ra cteristic of R . C o n seq u en tly , for any I eL , j

in d u ces a homotopy e q u iv a le n c e j : holim ° ( Z A ) J ( Z . ) e t) - holim o ( Z / £ ) J ( X . ) e t) . ■

We conclude this ch ap ter with the following s p e c ific com parison theorem which has proved useful in many ap p licatio n s (s e e , for exam ple, the d iscu ssio n of C hapter 9 ).

The reader can co n su lt [40] for an e x p licit

d iscu ssio n of G ^ and its cohom ological p roperties. PROPOSITION 8 .8 . G

L e t G(C) b e a co m p lex red u c tiv e L ie group, a nd let

be an a s s o c ia t e d C h ev a lley in tegra l group s c h e m e . L e t F

the a lg eb ra ic clo s u re of the prime fie ld F p , of F

let R

den o te

b e the Witt v ecto rs

(a co m p lete d is c r e t e valuation ring with re s id u e fie ld F ), and let

R -» C b e a ch o s e n em b ed d ing. containing F

T h en for any a lg eb ra ic a lly c lo s e d fie ld k

the b a s e ch a n g e maps BG^ -> BG p -> B G R et -> Sing.(B G (C )) w hose pro-L L = P -{p }.

co m p letio n s a re w eak e q u iv a le n c e s in pro-H^ , w here C o n seq u en tly , for any prime £ / p ,

th e s e maps d eterm in e a

chain of homotopy e q u iv a le n c e s betw een h o U m o (Z /l)oo((B G k )e t)

and

(Z /E^oSing.C B G C C )) .

P ro o f. The fact that the b ase change maps isom orphisms in Z /m

Gk -» Gp -> GR B G F -» B G R (B G R)et S in g.(B G (C )) have

pro-L com pletions which are

weak eq u ivalen ces is given by Theorem 8 .4 (and the homotopy eq uivalen ce Sing.(|BG^?P|)->Sing.(BG(C)) ). P roposition 6 .1 0 and C orollary 7 .2 now imply that the chain of maps ( 8 .8 .1 ) determ ines a chain of homotopy eq u iv alen ces relatin g holim ° ( Z / £ ) 00((B G k)e j.) and

(Z /£ ) oo°S in g .(B G (C )).

9.

APPLICA TIO N S TO TO POLOGY

In th is ch ap ter, we co n sid er two c la s s e s of top o lo g ical ap plications to the theory we have developed.

In Theorem 9 .1 , we present (a modified

version o f ) D. S ullivan ’s proof of the Adams C o njectu re.

T his is followed

by an infinite loop s p a c e version of the Adams C onjecture (Theorem 9 .2 ) whose proof requires the rigidity of the e ta le to p o lo g ical type and the B ousfield -K an co n stru ctio n s.

Theorem 9 .3 p resen ts a method of c o n stru c­

tion of maps of lo calized cla ss ify in g s p a c e s of L ie groups which are not induced by homomorphisms, w hereas Theorem 9 .5 d escrib es a re la tiv iz a tion of th is con stru ctio n to homogeneous s p a c e s .

T h ese co n stru ctio n s

u tilize p ositiv e c h a ra c te ris tic alg eb raic geometry. The reader will observe that a ll of the ap p licatio n s of th is chapter involve the study of sim p licial sch em es. In studying the sta b le homotopy groups of sp h eres, one employs the J ‘homomorphism J : O -» O ^ S 00 given by sending a : Rn -» Rn in On to the restrictio n of a ,

J ( a ) : S n_1 -> Sn_1

in Qn“ 1Sn -1 . In a se rie s of

papers, J . F . Adams determined the order of the image of J * : 77^(0) -> 00S°°) = tt^ (S °) up to 2-to rsio n [2].

While in v estig atin g this

J-homomorphism, Adams was led in [1] to a dram atic co n jectu re (the “ Adams C o n jectu re ” verified in Theorem 9 .1 ) concerning the generalized J-homomorphism con sid ered by M. F . Atiyah J : KO(X) -> J ( X ) (sending a real v ecto r bundle to its a s s o c ia te d sp h erical fibration) for a finite com ­ plex

X

[9], w hose solution for sp h eres com p letes the determ ination of

im (J* )C 4 ( s ° ) . In an influential paper [61], D. Quillen outlined a proof of the com plex K-theory analogue of the Adams C onjectu re.

T h is outline employed e ta le

homotopy theory, thus su ggestin g that the formalism introduced by M. Artin 80

81

9. APPLICATIONS TO TOPOLOGY

and B . Mazur to study a b s tra ct alg eb raic v a rie tie s might be “ turned around” so a s to u tilize alg eb raic geometry in the study of alg eb raic topology.

D. Q u illen ’s outline w as com pleted by the author in his th e sis

(cf. [29]); prior to the publication of th is com plete proof, Quillen in [63] provided an entirely different proof of the Adams C onjectu re based on the e arlier work of Adams and a technique of “ approxim ating” the orthogonal group by finite groups (s e e Chapter 1 2).

Independently, D. Sullivan pro­

vided a proof of the Adams C onjectu re which a ls o used e ta le homotopy theory ([6 9 ]); this proof differed in many re s p e c ts from Q u illen ’s outline, e sp e cia lly in that it does not involve alg eb raic v a rie tie s in c h a ra c te ris tic p > 0. We begin with an outline of S u llivan ’s proof of the Adams C onjectu re. The reader should re ca ll that the Adams operation

9*9 on (re a l) K-theory

KO(X) is determined by sending a bundle E -> X to the q-th Newton polynomial in the e x te rio r powers

A^E -> X

of E

X

(so that ^ ( E ->X)

is represented by a virtual bundle). T H E O R E M 9 .1 (Adams C onjectu re).

L e t J : BSO -» BSG re p r e s e n t the

(rea l) J -homomorphism, w here SO = USO(n)

is the in fin ite s p e c ia l

orthogonal group and w here SG = USGn with SGn the monoid of o rien ted s e lf-e q u iv a le n c e s of Sn . L e t ¥ 9 : BSO -> BSO re p re s e n t the q-th Adam s operation on (rea l) a lg e b ra ic K -theory, som e q > 0 . Jo ¥ 9 ,

T h en

J : BSO -> BSG

d eterm ine hom otopic maps of Z [ l / q ]

lo ca liza tio n s

Jo1*q ~ J : ( B S O ) 1 / q - ( B S G ) 1/q .

P roof (sk e tch ).

B ecau se

homotopy groups,

BSG is a simply co n nected s p a ce with finite

(B S G ). .

is homotopy equivalent to

\ where (

= holim1Su( ) ° (

II

(BSG)p,

ir* )£ . B e c a u s e

facto r through BSO -> (B SO )£,

J °*Pq , J : BSO -> BSG - (BSG)'^'

it su ffice s to prove that

82

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

J o ip q ,

are equal in

J : (BSO)g - (BSG)g

. We re c a ll that there e x is ts a g alois automorphism

o e G al(C ,Q ) su ch that

■An = V ° e t '

;

- ((BSOn ,c)et^ - « BSOn,C)et)'g - (BSO(n))'£

s ta b iliz e s (with re s p e c t to n ) to determine

^

: (BSO)£ -> (B S O )p

6n is determined by the two right-m ost arrows of P rop osition 8 .8 .

where C o n se ­

quently, it su ffice s to prove that J o*An > J :( B S O ( n ) ) - - .( B S G n_ 1) are equal in

for e a ch

n > 0.

One v erifies th at maps in

into (BSGn_ 1)^' are equivalent to fibre

homotopy c l a s s e s of fibrations with fibres determ ines the

(Sn - 1 ) p

where X -> (B S G ^ P g

(Sn _1)~ fibration given by the pull-back of the universal

fibration B ((S n“ 1)p (S G n l )p -» B ((SG n_ 1) p .

The fa c t that the following

square B S O (n -l) -------------- - B (S n - 1 ,SGn_ 1)

i

BSO(n) ----------- - --------- - BSG n- i is homotopy ca rte s ia n im plies that

i : (B S O (n -l))£ -» (BSO(n))^ corresponds

to J : (BSO (n))£ -» ( B S G ^ P g . Moreover,

^An_ 1 and \fin fit in a homotopy

comm utative square iff (B S O (n -l » £ -----------

i

(BSO(n))^'

(B S O (n -l))£

i

(BSO(n))g

;

83

9. APPLICATIONS TO TOPOLOGY

b ecau se the horizontal arrows of this square are homotopy e q u iv alen ces, th is square is n e c e s s a rily homotopy c a rte s ia n .

We conclude that and J °ipn for any

i : (B SO (n-l))^' -» (BSO(n))^ corresponds to both J n>0,

so that J = J

in

. ■

We re ca ll th at an 0,-spectrum X and maps

in : 2 X n -> X n+1

a homotopy eq u iv alen ce

is a seq u en ce of pointed s p a c e s

for n > 0 su ch that the adjoint of e a ch

i n is

X n -> ^ X n+1 . The J-homomorphism J : BSO ->

BSG exten d s to a map of (co n n ected ) O -sp ectra Adams operation ^

Xn

J : bSO -» BSG and the

: BSO -» BSO determ ines a map of O -spectra

: (bSO)1 ^ -> (bSO )1 ^ . It is natural to ask whether the maps J , J o ^ q : (bSO)x /q -> (B S G )1 ^q are homotopic as maps of sp e c tra . fortunately, this is not the c a s e :

J , J

homotopic as maps of H -s p a c e s , so that J

: (B S O )1 ^ -> (B S G )^ and J

Un­ are not

cannot have

homotopic restrictio n s to the first delooping of (B S O )1//q ([5 6 ]). N on eth eless, the proof of Theorem 9.1 exten d s to the following spectrum (or “ infinite loop s p a c e ” ) version of the com plex Adams Con­ jectu re proved by the author in [37].

The proof proceeds by carefully

refining S u llivan ’s homotopy th eo retic arguments s o that they remain valid in the much more rigid co n te x t of sp e c tra .

The key step is to interpret a

homotopy c la s s of maps of sp e ctra into (Z /f ^ ^ B S G ) in terms of a geom etric stru ctu re which a ris e s from alg eb raic geometry. T H E O R E M 9 .2 .

L e t J : bU -» BSG

of co n n e c te d £l-spectra and

d en o te the ( co m p lex) J-homomorphism

: (b U ^ ^ ^ (b U )j ^

d en o te the q-th Adam s

operation on the sp ectru m (b U )1 / q for som e q > 0 .

Jo 1 , where ?

n = l0 ,** * ,n },

is the categ o ry of finite pointed s e t s ,

p- : n -> 1 s a tis f ie s

p^(i) = 1 , P j(j) = 0. for j ^ i .

be obtained by ca te g o rica l co n stru ctio n s: BSG a rise from 3~-spaces. J,

and

Such “ ^ - s p a c e s ” can

in p articu lar, both bU and

The proof proceeds by verifying that

J o ¥ q : ( Z /£ ) ooo b U . ( Z / f ) oooBSG

are “ hom otopic” maps of 3"-s p a c e s for any prime Homotopy c l a s s e s of maps of 3"-s p a c e s

B -> ( Z / l ) ^ ° BSG are se e n

to be in natural one-to-one correspon d ence with S2 -fibrations ” over B .

“ Z /£-com pleted

The p re cise definition of su ch a structure

([3 7 ], Definition 7 .2 ) is quite su b tle, having been modified repeatedly to permit such a c la s s ific a tio n theorem for maps into ( Z /f ) ^ ° B S G . The pull-back of the universal via the map J

Z /f-com p leted

S2-fibration over ( Z / l ) ^ ° BSG

is represented by a structure arisin g from algeb raic

geometry (an elab o rate version of holim ° ( Z / £ ) oo °B (A ^ - { o ! ,( G L n £ ) ) e j- -> holim ° ( Z /£ ) oo°B (G L n ^ )et which refines the com plex analogue of (B S O (n -l))^ -» (BSO(n))g in the proof of Theorem 9 .1 ). The map ^

: (Z /Q ^ o b U -> ( Z / Q ^ b U

corresponds to a galois action

of the alg eb raic geometry model (an elab orate version of ae t : holim ° ( Z /E ) ^ ° B ( G L n^C)e t - holim ° ( Z / e ) 0O° B (G L n>c)et )

and is thus covered by a map of

Z /£-com pleted

S 2 -fibrations.

quently, the c la s s ific a tio n theorem implies that J

a nd J

C o n se ­ are

homotopic maps of 3"-s p a c e s into ( Z /f ) ^ ° BSG . ■ D. Quillen once su g g ested that the co n stru ctiv e a s p e c t of algeb raic geometry could provide a valuable approach to various e x is te n c e problems in alg eb raic topology.

Theorem s 9 .3 and 9 .5 (described below) are

exam ples of the s u c c e s s fu l ap plication of th is philosophy.

85

9. APPLICATIONS TO TOPOLOGY

We first con sid er the problem of co n stru ctin g a map between (lo ca liz e d ) c la ssify in g s p a c e s of com p act, con nected L ie groups which is not the cla ssify in g map of a homomorphism. ch aracterized th o se maps B G

In [4], J . F . Adams and Z . Mahmud

> BG which could be “ defined after

finite lo c a liz a tio n ’ ’ in term s of “ a d m issib le ” maps between the u niversal covering s p a c e s of the a s s o c ia te d maximal to ri.

The lo ca liz a tio n n e c e s ­

sary before Adams and Mahmud were assu red of the e x is te n c e of a map on cla ssify in g s p a c e s included the inversion of all primes occurring in the Weyl groups of G and G '.

C onsequently, the following e x is te n c e theorem

provides a sig n ifican t sharpening of one a s p e c t of the work of Adams and Mahmud. One in terestin g a s p e c t of th is theorem is its u se of c h a ra c te r is tic

p

alg eb raic geometry (a s did the proof of the com plex Adams C onjectu re given in [29]).

The theorem its e lf is stated without proof in [36], and the

proof given is an e a sy g en eralization of that given in [33] for a le s s general co n te x t. T H E O R E M 9 .3 .

L e t G(C) an d G '(C )

group s and let f : GF -> Gp

be a homomorphism of a s s o c ia t e d a lg eb ra ic

gro ups over S p e c F , w h ere F F .

he co m p lex re d u c tiv e a lg e b ra ic

is the a lg e b ra ic c lo s u re of the prim e fie ld

T h en f and a c h o ic e of em bedding of the Witt v ecto rs of F

(cf . P roposition 8 .8 ) determ in e a map $ : ( B G ( C ) ) 1 /p - ( B G '( C ) ) 1 /p fitting in a homotopy com m utative sq u a re

B^. is the lifting of the restrictio n of f : Gp -> Gp of GF

to maximal tori

and G p .

P roof (Sketch).

A ch o ice of isomorphisms Tp

^

G L^r^

and

T p ^ G L * r^ determ ines an isomorphism between the group of r x i integer-valued m atrices and the group of alg eb raic group homomorphisms from Tp to

T p . T hu s, we observe that the re strictio n of f ,

: Tp -» T p

fits in a com m utative diagram of group sch em es

tf

TC

Tp-------------

T£,

where R denotes the Witt v e cto rs of F

and where T^, -> TR , T£, -» T p

are determined by the ch osen embedding R -> C . By Prop osition 8 .8 ,

f induces a map

(B G (C ))£ -> (B G ^C ))^

restrictin g to (B (B T '(C ))£ for any holim Su° (

/ .

B ecau se

I ^ p,

where

( )£ =

H *(B G (C ),Q )® Q j = H *((B G (C ))g',Q) and

H *(B G /(C ),Q )® Q g = H * ((B G '(C ))p Q ),

we conclude that (B ( B G ^ C ) ^ ^ finite sk eleta of (B G (C ))1 ^/p .

whose homotopy c la s s is well defined on T o prove the uniqueness (up to homotopy)

of th is map O , we employ the Milnor e x a c t seq uen ce * ^ W lH o m j^ S s k ^ C B G C C )) ! /p ),(B G X C ))1 /p )j ■ ^ { H o n i j , (s k ^ B G C C ))!

We re ca ll that H -((B G (C ))1 is even.

),(B G X C ))1

)!-* .

and 77-|((BG/(C ))1 ^ )

are finite u nless

i

T h erefore, obstruction theory im plies that each of the groups

Homj^ ( 2 s k n(B G (C ))1 ^ , (B G /(C ))1 ^ ) zero .

((BC C C )^ /p ,(B G '(C ))1

is fin ite, so that the

lin^-term is

T his im plies the uniqueness of O . ■

As a corollary of Theorem 9 .3 , we obtain the following e x ce p tio n a l eq u ivalen ces exhibited by the author in [3 3 ], the la s t of which were first exhibited by C . Wilkerson in [71].

T h e se eq u iv alen ces are th o se d eter­

mined by the 11 ex c ep tio n a l is o g e n i e s ” of alg eb raic groups. COROLLARY 9.4. ( B S 0 2n^ 1) 1

a.)

T h e re e x is t s a homotopy e q u iv a len ce

-> (BSpn) 1//2

for any n > 0 .

eq u iv a len ce $ : (B G 2) 1 ^ -> (B G 2) 1 ^

b .)

T h e re e x is t s a s e l f

w hich re s tric ts to B0|. : B T 2 ->BT2 ,

w here ^. s e n d s a short root to a long root an d a long root to 3 tim es a short root,

c .)

T h e re e x is t s a s e l f e q u iv a le n ce $ : ( B F ^ ^ ^ (^ ^ 4) 1/2

w hich re s tric ts to B0^. : B T 4 ->BT4 , w here

s e n d s short roots to long

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

88

roots and long roots to 2 tim es short ro o ts. d u ctiv e L i e group and any prime p > 0 , O : (B G (C ))1 /p - (BGCC))! /p

d .)

For any co m p lex r e ­

there e x is t s a s e l f e q u iv a le n c e

s u c h that 0 * : H2n(B G (C ),Q ) - H2n(B G (C ),Q )

is m ultiplication by pn for e a c h n > 0 . ■ The com parison theorem s of Chapter 8 immediately imply that g alois a ctio n s determ ine s e lf eq u iv alen ces on Z /£-com pletion s (but not n e c e s ­ sarily on ration al cohom ology).

It would be in terestin g to understand how

the se lf eq u iv alen ces of ( Z /£ ) oo°S in g .(B G (C )) determined by G a l(F ,F p ) as in P rop osition 8 .8 depend on the ch o ice of embedding of the Witt v e cto rs into

C.

R ecen t work of Z . Wojktowiak [72] appears to answ er

this question for G(C) = G L n(C ) and I > n . In the following theorem , we present a re la tiv izatio n of Theorem 9 .3 concerning h o m o gen eo u s s p a c e s which w as proved by the author in [36]. THEOREM 9 .5 .

L e t G = G (C ), G ' = G '(C )

g ro u p s; let G ^ , G^ a n d let Spec Z .

Fp,

be a s s o c ia t e d C h ev a lley group s c h e m e s over Spec Z ;

C G^ , H^ C G^ L e t f : Gp -» Gp

groups over S p e c F ,

b e co m p lex red u ctiv e L ie

be c lo s e d su bgro u p s c h e m e s red u ctiv e over

b e a homomorphism of a s s o c ia t e d a lg eb ra ic

w here F

is the a lg eb ra ic c lo s u re of the prime fie ld

s u ch that f re s tric ts to f| ‘ Hp -> Hp .

em bedding of the Witt v ecto rs of F

T h en f ,

f| , and a c h o ic e of

into C naturally d eterm in e a homotopy

c la s s of maps ® :( G /H ) i / p - ( G 7 H ') 1 / p .

P roof (Sk etch ).

The map 0 ^

: ( G / H ) ^ -» ( G '/ H ') ^

is defined as the

unique (up to homotopy—cf. [36]) map fitting in a “ map of fibre tr ip le s ” (G /H )(0 ) --------------- - (B H )( 0 ) --------------- - (B G )(0)

(G '/H ')( 0 ) --------------- -- (B H ')( 0 ) ---------------

(B G O (0)

9. APPLICATIONS TO TOPOLOGY

89

whose middle and right v e rtica l arrows are obtained a s in Theorem 9 .3 . We re call that G /H

is naturally homotopy eq uivalent to B (G ,H ,* ). More­

over, the proof of P rop osition 8 .8 ap plies to prove that each map of the chain (determined by a c h o ice of embedding of the Witt v e cto rs (B (G F ,H F ,* ) e t) ^

R

into C )

(B (G R ,H R , * ) e t) ^ ( B ( G c ,H c ,* ) e t)£' (B (G c ,H c ,* ) s>et)£ -»(Sing. (B (G ,H , *)))'£

is induced by a weak eq u iv alen ce in pro-K^ and is thus a homotopy eq u iv alen ce, where

( )£ = holim S u ° ( / .

The map

(G /H )£ -> (G '/H ')£

is that induced by B (f, th ese eq u iv alen ce s. The com m utativity in

(g /h)(o>—*

of the following squares

n«G/H)p(o) ^ U

o

m -

m>~

( 0) £ £

(G7Ho(0)

U p

p

----- U ^ V io r----£/p

n(Q'/H'), Up

follow s from the uniqueness of O p ( ( G / H ) p ^ -> ( ( G '/ H 'J p ^ “ map of fibre tr ip le s .”

fitting in a

C onsequently, the finite dim ensionality of G/H

implies that th e se sq uares determine a unique homotopy c l a s s of maps V ..

s a tis fy the condition that g be a map of

rigid hypercoverings over f and w hose maps are com m utative sq uares of rigid hypercoverings over f . Define H R (f) whose o b je cts

g : U . . -» V ..

S p (f) to be the full su bcategory of

sa tis fy the condition that U .. -> V .. x y

be sp e c ia l (cf. proof of Prop osition 3 .4 ). Define the homotopy fib re s fib(fj1^_), fib(fe t) , and fib(fgp) by

fib (fht)

= l ( 7 r ( A g ) ' ) - 1(y ) ; g e H R ( f )S

fib (fe t )

= K ^ A g f r ^ y ) ; g e H R R (f )i

fib(fs p ) = i(» U .. -*V.. x y> X . ). We define ^ r':H R (f)

g : U . . -» V ..

to

V .. ,

95

10. COMPARISON OF FIB R E S

s o th at there are ca n o n ica l isomorphisms ^

n (A r(g ))~

in H J

and ( F ( A ^ ( g ) ) 'r 1(y ) ^

( tK A ^ X b ) ) ' ) - 1 ^ )

^

H* .

fib(f^t) via the functor x\s and the natural maps

We define fib(fg p)

MA^g))*)-1^) ^ (rr(Ar(g))'r\y) — (r7-(Ag)~r 1(y). T he reader can e a s ily verify that fib(fg p) -» fib(fj1^.) is homotopy inverse to fib(fht) "* fib(fsp ) nsing the fa ct that

g -> ^r(g) -♦ (9(g) and g -> \Jj '(g ) -» 0(g)

are homotopic. T o verify the secon d a sse rtio n , we observe that the com position (X . x Y> y )ht -> fib(fht)

fib( fsp)

is given by the functor i* ° s : Sp(f ) -*H R (X . x y > y ,y )

sending

g : U . . -» V ..

to

U .. x y y . The proof of the a ss e rtio n follow s

from the e x is te n c e of natural in clu sion s HR(X. x y y , y)

U .. x v

v -* U .. x y y in

determ ining a natural transform ation

rj -> i * ° s . ■

The next proposition exp lain s our introduction of Sp(f ) and our co n ­ sid eration of fib(f s p ) .

The proposition originally appeared as P rop osition

2 .4 of [29]. PROPOSITION 1 0 .4 .

L e t f : X -» Y

s c h e m e s and let fjTp(y) £ Pro'^ *

fsp1^ ) = ^77(u - x v.

b e a p o in ted map of lo ca lly n o etherian

b e d e fin e d by

g : U - ^ v - in s P(f )! = S ^ Cg r Hy ) ; g f S p ( f ) i

96

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

(w here vQ -» V.

is the in clu sio n of the d is tin g u is h e d com ponent of VQ ).

For any locally co n sta n t a b elia n s h e a f M on E t ( X ) ,

th ere e x is t s a

natural isom orphism

H*(f s p ( y ) ’ M)

(R *f*M )y

w hich fits in a com m utative sq u a re

^ (R*f*M)y

H *((X y )h t,i*M ) -------—

-------

H *(X y ,i*M )

w here i : X y = X x y y h>X is the g eo m etric fib re of f ,

the left arrow is

in d u ced by the map (X y )^ -> f^"p(y) g iv e n by rj of P roposition 1 0 .3 , the right arrow is the ca n o n ica l b a s e ch a n g e map, a n d the bottom arrow is the isom orphism of P roposition 5 .9 . P roof. The functor

77: Sp(f )

HR(Xy,y) of P rop osition 1 0 .3 induces the

following map of s p e c tra l se q u e n ce s (whose co lim its are indexed by g : U.

V.

in S p (f) and W.

in HR(Xy,y) ) b e ca u s e

Up x y

vQ -» X x y vQ

is a hypercovering

E j '^ = colim H^(Up x v

'E ^ ’q =

vQ,M) = > colim HP+9(U . x y vQ,M)

colim Hq(Wp,i*M )

=> colim HP+C1(W. ,i*M ) .

By P rop osition 3 .7 , colim H*(U. x y vQ,M) = colim H *(X x y vQ,M) = (R*f^M)y and colim H*(W. ,i*M ) = H *(X y ,i*M ) .

10. COMPARISON OF FIB R E S

97

X y -» X x y vQ

The map on abutments is induced by the clo se d immersions and is therefore the b a se change map.

As se e n in Theorem 3 .8 , the

sp e c tra l seq u en ce c o lla p s e s at the E 2 -le v e l with E * > ° ^ H * ( ( X y )h t,i*M )

a s in P rop osition 5 .9 .

Sim ilarly, the

E^^

s p e c tra l se q u e n ce s c o lla p s e s

at the E 2-lev el (use the proof of Theorem 3 .8 with the presheaf G3 rep laced by H^( ,M) ) and E 2 ' ° = colim H*(77(U. xv v ),M )). L e t H *(fSp1(y),M )

(R*f^M)y be the edge isomorphism

E *’0

E ^ . Then

the a sse rte d com m utative diagram is a co n seq u en ce of the naturality of the edge homomorphism. ■ Although P rop osition 1 0 .4 has been presented only for maps of sch em es, its ap p licab ility to maps of sim p licial sch em es is a con sequ en ce of the following lemma. L E M M A 1 0 .5 .

L e t f : X . -> Y .

b e a po in ted map of lo ca lly n o etherian

sim p licia l s c h e m e s with the property that e a c h X R a nd ea c h Yn is c o n ­ n e c t e d , and let M b e a lo ca lly co nstant a b elia n s h e a f on E t ( X .) . map (X . Xy y \ t -» fib(f ea ch

If the

) of P roposition 1 0 .3 in d u ces isom orphism s for

n > 0 H *(fib ((fn)s p ),Mn) ^

H *((X n x y

y )h t,i*M n) ,

then this map a lso in d u c es an isom orphism H *(fib(fsp ),M) ^

Proof. F o r any g : U . . -»V ..

H *((X . xy> y )h t,i*M ) .

in H R (f),

defined by the condition that ( 77(g) ) for each

n > 0.

Then

A ( 77(g)~) -» 77(A g )~ : sim p licial s e ts

n(Ag)

let 77(g) :77(U ..)

tt^ V ..)

be

be the mapping fibration of ^(gn)

77(Ag)~ fa cto rs through a natural map

this is proved by observing for any map of bi-

h :S .. -> T ..

and any

: A[n] -> A[n] (the inclusion of

s k J1_ 1A[n] minus the i-th fa c e into A[n] ) th at a map e-

hn

naturally

98

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

determ ines a map a^ -» A (h ). Moreover, b e ca u s e con nected for each

n > 0,

77(Un ) and 7r(Vn ) are

Theorem B .4 of [12] im plies that

A w g y r ^ y ) - > ( ^ ( A g ) 'r 1(y ) is a weak eq u iv alen ce. C onsequently, the natural map f Z

is a pointed

fa cto rs through g ,

then

7 0 s in c e

colim H^(v x z Z ',M ) = 0

for

q > 0

y->v

and any geom etric point y of Z rep laced by H^(

(cf. proof of Theorem 3 .8 with G3

x z Z ',M ) ). ■

We now present our b a sic com parison theorem , a g en eralizatio n to sim p licial sch em es of [2 9 ], Theorem 4 .5 . THEOREM 1 0 .7 . su ch that ea ch ea ch

77'1((Yn)j1^;)

L e t f : X . -* Y .

b e a p o in ted map of sim p licia l s c h e m e s

X n is c o n n e c te d , ea c h Yn is c o n n e c te d and no eth eria n , is p ro fin ite, a n d n 0(X . x y y )

lo cally co n sta n t a b elia n s h e a f on E t(X .)

is fin ite . L e t M be a

s u c h that for e a ch

n,q > 0 ,

R (fn)” i ( y ) ”

(R qfn*Mn)v H*f(Xn x y y,i*M n) implies that y n induces an isomorphism

H *((fn)sJ(y)> Mn) ^ by P rop osition 1 0 .4 .

H* « X n > 0 *

We let C denote the left d irected categ o ry of co n n ected , pointed, p rin ci­ pal G -fibrations of fn by Y n

Y ^ n> Y n , and we le t f ^ : Yn • T he maPs

(f n)Sp

-> X n denote the pull-back

(fn)sp

indexed by

-» Yn in C

determ ine maps of sp e ctra l seq u e n ce s inducing isomorphisms on abutments

E 2*q(Yn

Yn) =

HPCCY^t.H'kfibCf^.M)) => H ^ X ^ M )

' E P ^ Y ; ^ Yn) = H P((Yn)h t, H'l((f n)3 J(y ),M )) = > HP+cl(X n,M) . The hypothesis that R ^ n5ie'Mn is lo ca lly co n sta n t on Et(Y n) im plies that R ^ f^ M n (the re strictio n of the fa ct th at

) is lo cally co n stan t on E t ( Y ^ ) ;

' e P, c* ca n be identified with the L eray s p e ctra l seq uen ce

im plies that the c o e fficie n ts for the cohomology groups of (Y ^ )j^ lo ca l co efficien t system in the c a s e of

are a

^ a s we^ as *n the c a s e °f

. (We have im plicitly used the noetherian hypothesis on Yn to obtain the E 2-term s in the above form by taking a d irect limit of c o e ffi­ c ie n ts ; for more d e ta ils , s e e [29], 4 .3 .) The map f^ -> fn cle a rly induces an isomorphism on geom etric fibres x n xY; y ' ^ n

x n xY y • n

M oreover, Lemma 1 0 .6 im plies that fib(f^) -» fib(fn) is an isomorphism in p ro -K ^ . We tak e the colim it (indexed by C °P ) of the maps

E P ,q (Y ; - Y n) s o that colim

' E P ' ^ y ; - Y n)

E ^ ’^ is cohomology with triv ial c o e fficie n ts (the a ctio n

of „ (Yn) on R Yn ); moreover, colim E V *

is a ls o cohomology with triv ial co e fficie n ts b e ca u s e

77i((Y n)j1j:)

is pro-finite and the a ctio n of 77i((Y n)]l t ) on H *(fib(fn),M) is continuous.

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

102

Comparing th e se colim it sp e ctra l se q u e n ce s, we conclude that the map H *(fib ((fn)s p ),M) ^

colim H * ( f i b ( ( f ^ p),M)

— colim H * ( (Q -l (y ),M ) ^

H *((fn)-J(y ),M )

is an isomorphism a s required. The proper smooth b a se ch ange theorem ([7 ], X V I.2 .2 ) a s s e rts that f : X -» Y

and M e AbSh(X) a s below s a tis fy the h ypotheses of Theorem 1 0 .7 .

C O R O L L A R Y 1 0 .8 .

L e t f : X -> Y

b e a p ro p er, sm ooth, po in ted map of

c o n n ected , no eth eria n s c h e m e s and let M be a lo ca lly co n sta n t, c o n ­ stru ctib le s h e a f on E t(X )

with sta lk s of order rela tiv ely prim e to the

re s id u e c h a ra c te ris tic s of Y .

If ^ 0 ^ )

is pro-finite, then the natural

map (Xy >et -> fib(fe t ) in d u c e s an isom orphism H *(fib(fe t ),M) ^

H *((X y )e t ,M) . ■

In the proof of the Adams C onjectu re presented in [2 9 ], the following s p e cia l c a s e of Theorem 1 0 .7 was required (cf. [2 9 ], 5 .3 ). proof that fib(f^t) is weakly

An independent

£-equivalent to a sphere has been pro­

vided by D. C ox in [21] (using the e x is te n c e of a “ Thom isom orphism ” ). The fa ct that the map f : X -> Y

and the ab elian group of C orollary 1 0 .9

sa tisfy the hypotheses of Theorem 1 0 .7 is verified in P rop osition 5 .2 of [29]. C O R O L L A R Y 1 0 .9 .

L et Y

b e a pointed, c o n n e c te d , g eo m etrica lly u ni­

b ra n ch ed n oetheria n s c h e m e and let E on Y .

L e t V (E ) = Sym (Ev)

be a co h eren t, lo ca lly fr e e s h e a f

be d e fin e d lo ca lly as the sp ectru m of the

sym m etric a lgeb ra of the dual of E

over Oy

and let f : X -> Y

the stru ctu re map V ( E ) - o ( Y ) -> Y , w here o : Y -» V (E ) For any fin ite a b elia n group A c h a ra c te ris tic s of Y ,

d en o te

is the 0 -se c tio n .

of order rela tiv ely prim e to the re s id u e

the natural map (X y )et -> fib(fe t)

isom orphism in cohom ology H *(fib(fe t),A ) ^

H *((X y )e t ,A ) . ■

in d u ce s an

11.

A PPLICA TIONS TO GEOMETRY

In th is ch ap ter, we present four ap p licatio n s of e ta le homotopy theory to geom etry.

Theorem 1 1 .1 is a resu lt of P . Deligne and D. Sullivan

a sse rtin g that a “ f l a t ” v ecto r bundle ca n be triv ialized by p assin g to a finite covering s p a c e of the b a se .

T he next ap p licatio n , C orollary 1 1 .3 ,

is a n e ce ssa ry and su fficien t condition due to M. Artin and J . -L . Verdier for a real alg eb raic variety to have a real point; the proof we present is based on a resu lt of D. Cox concerning the e ta le homotopy type of a real v ariety.

Theorem 1 1 ,5 p resen ts a long e x a c t homotopy se q u en ce a s s o c i ­

ated to certain maps ( “ geom etric fib ratio n s” ) of sch em es and their geom etric fib res.

F in a lly , we verify in Theorem 1 1 .7 that a smooth a lg e ­

braic variety over an alg e b ra ica lly clo se d field

k has a b ase of e ta le

neighborhoods w hose e ta le homotopy types have 2-com pletions which are K(77,1 ) ?s

for £ a prime invertible in k .

We re c a ll that a com plex v ecto r bundle over a pointed, connected b ase has a d iscre te stru ctu re group if and only if it a ris e s from a com plex rep resen tation of the fundamental group of the b a se (a s the bundle a s s o c i ­ ated to the corresponding lo ca l co e fficie n t sy stem whose fibres are com ­ plex v ecto r s p a c e s ).

If the b a se of a finite dim ensional, com plex v ecto r

bundle is a manifold, then the bundle has d isc re te stru ctu re group if and only if the bundle adm its an integrable con nection (in which c a s e , the bundle is said to be “ f/a f” ).

The co n clu sio n of the following theorem

describ in g how su ch a v e cto r bundle can be triv ia liz e d (but not ad dressin g the q uestion of triv ializin g the a s s o c ia te d rep resen tation of the funda­ mental group of the b a s e ) is a homotopy th e o re tic statem en t amenable to e ta le homotopy th eo retic tech n iq u es.

103

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

104

T h e o r e m 1 1 .1 (Deligne-Sullivan [25]). com plex and let p : E -> T

b e a fin ite sim p licia l

b e a fin ite d im en sio n a l, co m p lex v ecto r bundle

with d is c r e te stru ctu re gro u p . sp a ce T '-> T

L et T

T h en th ere is a fin ite, s u rje c tiv e co v erin g

s u c h that the p u ll-b a ck p ' : E ' - > T ' of p to T '

is a

trivial v ecto r b u n d le. P roof (S k etch ). con n ected .

We are e a sily reduced to the c a s e

is pointed and

p : 7r1(T ,t) -> G L n(C ) be a rep resen tation determining

Let

p : E -» T . B e c a u s e

77'1(T ,t)

is finitely gen erated , we may find a subring Z su ch th at p is given by p : zr1(T ,t) ->

A of C of finite type over G L n(A ).

T

D eligne and Sullivan show that the c la s s ific a tio n map for

p , f : T -> B U n , ca n be represented up to homotopy by a map g A( C ) : X A(C )t0 P - G ra s s N+n n . A(C )to P , where

g A : X A '* G r a s s N + n ,n ;A

is a map of sch em es defined over A . Let

m and

m' be maximal id eals of A whose (fin ite ) residue fields

A /m = k and A /m ' = k / have d istin ct c h a r a c te r is tic s .

An argument

sim ilar to th at of the proof of P rop osition 8 .8 v erifies that

gA has the

property th at (g A(C )to P /

(g p )^ . , where

L

is tic of k (re s p ., K 01

Let

T '^ T

(g ^

and (g A(C )to P /

~

L ' ) denotes the s e t of a ll primes e x ce p t the c h a ra c te r­

(re sp e ctiv e ly ,

triv ial if (g r )

~

k ' ).

T his im plies that (g A(C))^°P is hom otopically

and {g-j-,) K

0L

are hom otopically triv ia l.

be the fin ite, s u rje ctiv e , pointed covering sp a c e a s s o c i ­

ated to the subgroup (ker p^) fl (ker p ^) of zr1( T , t ) , where p 1 : tt^{T ,t)-^ G L n(k) and p [ : 7r1(T ,t) -> G L n(k r) are induced by p : 7r1( T ,t ) ^ G L n( A ) . Applying the above argument to p ' : ^ ( T ^ t ) -> G Ln( A ) , we conclude that the cla ssify in g map for p ' : E ' - > T ' is hom otopically triv ial. ■ The following result of D. Cox d escrib e s the etale homotopy type of a real algebraic variety in terms of its a s s o c ia t e d complex analytic variety.

11. APPLICATIONS TO GEOMETRY

Let X

PROPOSITION 1 1.2 (D. Cox [20]).

105

b e a c o n n e c te d re a l a lg e b ra ic

variety ( i . e . , a r e d u c e d , irred u cib le s c h e m e of fin ite type over Spec R ), and let X ^ P

d en o te the co m p lex analytic s p a c e of co m p lex points of X .

T h en th ere is a w eak homotopy e q u iv a le n c e of Artin-M azur co m p letio n s (w ith r e s p e c t to the s e t P

of a ll p rim es)

x et

w here G is the group

P roof.

Let

group G . and

Z /2

~ (X j?P x |EG|)P

a ctin g on

by com plex conjugation.

Xp = X x Spec C , so that X p -» X L Spec R L Let

U. = c o s k ? ( X p ) ; L

b ecau se

U. =* X

is g alo is with

x cosk^Pe c ^ (S p ec C) Spec R

770(coskQPe c ^ (S p e c C )) = BG , there is a natural homeomorphism |ut°p|

as

|EG| .

X

^

G

The proposition now follow s d irectly from P ro p o sitio n 8.1 and Theorem 8 .4 . ■ The following criterion w as first proved by M. Artin and J . -L . Verdier and published by D. Cox with a proof based on P rop osition 1 1 .2 .

We

present another proof based on th is proposition. C o r o l l a r y 1 1 .3 (D. Cox [20]).

L et X

variety of dim ensio n n .

has at le a s t one rea l point if and only

T h en X

b e a c o n n e c te d rea l a lg e b ra ic

if H*(Xe j.,Z /2 ) A 0 for som e i > 2n . P roof. map X

If X

has a real point, then th is point is a se c tio n for the structure

Spec R .

B ecause

i > 0 , we conclude that

H ^.(Spec R, Z /2 ) = H * ( B Z /2 ,Z /2 ) ^ 0 for any

H *t( X ,Z / 2 ) ^ 0 for any i > 2n w henever X

a real point. C o n v ersely , if X

has no real point, then G = Z /2

has

a c ts

freely on X 4° P , s o th at X *°P x |EG| is homotopy equivalent to X ^ ?P /G . t b Q L B e c a u s e X^?P/G is a com plex an a ly tic v ariety of com plex dimension n and

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

106

Hi (X ^ ? P /G ,Z /2 )

^ H \ X e V Z /2 ) by P rop osition 11 .2 , we con clu d e that

H1(X e ^ Z / 2 ) = 0 for all

i > 2n whenever X

has no real points. ■

In [48], C orollary X .1 .4 , A. Grothendieck proved the e x a c tn e s s of the following seq u en ce 771(X y ) - 7 r 1( X ) - 7 7 1( Y ) - l whenever f : X -> Y

is a proper, smooth pointed map of noetherian, normal

sch em es with con n ected geom etric fibre X y . T h is seq u en ce was g en eral­ ized by M. Raynaud based on n otes of Grothendieck ([4 8 ], X III.2 .9 and X III.4 .1 ) to the asse rtio n that the following seq u en ce is e x a c t ^ ( X y ) 1- - ffj(X )/N - ^ ( Y ) - 1

where N is the normal subgroup of ^ ( X ) N = ker(ker(771(X )

L

defined by

^ ( Y ) ) -> k e r ^ X )

771( Y ) ) L ) ,

is the s e t of primes com plem entary to the residu e c h a ra c te r is tic s of Y ,

and f : X -> Y

is the re strictio n of f : X -* Y a s above to X = X - D

D is a d ivisor in X

with normal c ro s s in g s over Y

with

(a s defined below ).

We proceed to extend this la tte r homotopy seq u en ce to higher homotopy groups, observing that L-com p letion is e s s e n tia l (for exam ple, the Kunneth Theorem fa ils com pletely for c o e fficie n ts not prime to residu e c h a ra c te r­ i s t ic s ).

The homotopy seq u en ce will be proved for “ geom etric fib ratio n s”

which we now define. DEFINITION 1 1 .4 .

A map f : X -> Y of sch e m es is said to be a sp e cia l

geom etric fibration if f is the re strictio n of a proper, smooth map f : X ->Y to X - T ,

where T

condition:

T

c- in X 1

is a clo se d subschem e of X

is the union of clo se d su bschem es

over Y

is smooth over Y

sa tisfy in g the following T- of pure codim ension

su ch that e a ch non-empty in te rse ctio n of pure codim ension

c-

T fl-flT j A1 s

+••• + c- . (If e ach xs

c- = 1 ,

107

11. APPLICATIONS TO GEOMETRY

then T

is said to be a d ivisor in X

More g en erally ,

f:X

ea ch

V j, f j : f _ 1 (Vj)

is said to be a g eo m etric fibration if Y

Y

a Z arisk i open covering

with normal c ro s s in g s over Y .) admits

{Vj->Yi su ch that the re strictio n of f above

V j,

is a s p e c ia l geom etric fibration.

A geom etric

fibration (of re la tiv e dim ension 1 ) is said to be an elem en ta ry fibration if its geom etric fibres are co n n ected , affine cu rv e s. ■ In the following theorem (Theorem 4 .2 of [3 1 ]), our con sid eration of lf(i) : X (i) -> Y(i)S is a formal means of p assin g to the ( Y (i)

of Y , and let

b e the p u ll-ba ck of f by Y (i) - Y .

T h en

Y

there e x is t s a long e x a c t homotopy s e q u e n c e

w here L

- ” n « X y )e t) -

" V !Y « e t *> - -

is the s e t of prim es com plem entary to the re s id u e c h a ra c te ris tic s

of Y . P roof (S k etch ).

By Lemma 1 0 .6 , the natural map fib (!f(i)e ^J) -» fib(fe t) is

a weak eq u ivalen ce. map fib (jf(i)e t !) L

B ecau se

{Y (i)

is simply co n n ected , the natural

fib ({f(i)^ .j) is a weak eq u iv alen ce.

T hu s, it su ffices

to prove the natural map (X y)e t -> fib(fe t) is a weak L -e q u iv a le n ce ( i .e ., s a tis f ie s conditions a) and b) of C orollary 6 .5 ). Let

K denote

ker(77‘1(X e t ) -> 7r1(Ye t) ) L . B e c a u s e the rem oval of

smooth clo sed su bschem es of pure codim ension g reater than 1 d oes not affe ct the fundamental group ([7 ], XVI. 3 .3 ), we may apply the above Raynaud e x a c t se q u en ce to con clu d e that

771((X y )^ .) ->K is su rje ctiv e

108

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

with kernel H .

One v erifies that H is cen tral in 771((X y )^ j.). The

homotopy seq u en ce im plies that H ',

the kernel of the su rje ctiv e map

77’1(fib(fe j.)L ) ->K is a ls o ce n tra l. L e t M be a lo ca lly co n sta n t, co n stru ctib le , L -to rsio n abelian sh eaf on X (i) w hose re strictio n to some L-prim ary g alo is exten sio n is co n stan t.

X (i) '-> X (i)

We verify below th at we may apply Theorem 1 0 .7 to

f(i) and

M , concluding that (X )e j. -> fib(fe t) induces isom orphism s ( 1 1 .5 .1 )

H *(fib (fe t ),N ) _

for any lo cal system

N of finite, L-prim ary ab elian groups on fib(fe ^)

induced from a rep resen tation of K . 1 0 .7 for f(i) and M,

H *((X y )e t ,N)

To verify the hypotheses of Theorem

we may assum e f(i) is a s p e c ia l geom etric fibra­

tion with re la tiv e co m p actificatio n

j : X (i) -* X ( i ) .

A s in the proof of the

proper, smooth b a se change theorem ([7 ], XVI. 2 .2 ), it su ffice s to prove that R ^f(i) M comm utes with arbitrary b ase change on Y (i)

for a ll q > 0 .

U sing the L era y s p e c tra l seq u en ce and the proper b ase change theorem for f ( i ) ,

we are reduced to proving that R^j^M commutes with arbitrary

b ase change on Y (i) for a ll

p > 0

in order to verify the h ypotheses of

Theorem 1 0 .7 for f(i) and M.

T his is verified by exam ining the stalk s of

RPj^M at a geom etric point x

of X ( i ) - X ( i ) , computed a s the limit of

HP(

,M) applied to deleted e ta le neighborhoods of x .

e ffe ct upon (R^j*M )

F in a lly , the

of arbitrary b a se ch ange on Y (i) is controlled

using the coh om ological purity theorem ([5 9 ], V I.5 .1 ) thanks to the follow ­ ing co n seq u en ce of Abhyankar’s Lemma ([4 8 ], X II.5 .5 ):

on a c y c lic

covering of a deleted neighborhood of x , M can be extended to a lo cally co n stan t sh eaf on som e sm ooth, re la tiv e (to Y (i) ) co m p actificatio n . L e t ((X y )^t )H - (X y )L s p a c e s a s s o c ia te d to

( 1 1 .5 .2 )

and (fib (fe t) L ) H^ f i b ( f et) L be the covering

H and

H'.

Then ( 1 1 .5 .1 ) im plies that

« X y )e t)H - W

e P V

11. APPLICATIONS TO GEOMETRY

induces an isomorphism in

Z /£ cohomology for e a ch

isomorphism of (ab elian ) fundamental groups. (and, thus,

109

le L

and thus an

The proof that (1 1 .5 .2 )

(X y )^ . -> fib(fe j.)L ) is a weak eq u iv alen ce is com pleted by

proving that

H

H ' a c ts triv ially on the

Z /2 cohomology of the universal

covering s p a c e s of (Xy)^j. and fib(fe t) L . T h is la s t statem en t is proved using the following to p o lo g ical resu lt ([3 1 ], Appendix):

if F -> E -> B

a fibre triple of co n n ected , pointed s p a c e s , then the actio n of the homology of the universal covering of F

is on

fa cto rs through an actio n of

77^ £ ) . ■ The geom etric b a sis for our la s t ap p licatio n , Theorem 1 1 .7 , is the following theorem of M. Artin. T H E O R E M 1 1 .6 (M. Artin [7], X I. 3 .3 ).

L et X

be a smooth a lg e b ra ic

variety of dim en sio n n over an a lg eb ra ic a lly c lo s e d fie ld k an d let x be a c lo s e d point of X . U of x

T h en th ere e x is t s a Z a ris k i open n eigh bo rh o o d

w hose stru ctu re map U ^ Spec k may b e fa cto red a s a co m p o si­

tion of elem enta ry fibrations : f U - Un

ft > Un_ 1 ------> ••• ------> U 1 ------> S p eck . ■

In the sp e c ia l c a s e in which

k equals

C , Theorem 1 1 .6 a s s e r ts

that X to P admits the stru ctu re of the to tal s p a c e of an iterated fibration with fibres which are co n n e cte d , noncom pact Riemann s u rfa c e s . B e c a u s e the latter are

K(7t ,1 ) , s with n a free group, the homotopy seq u en ce im­

p lies that X to P is a ls o a

K (77,l) with n a s u c c e s s iv e exten sio n of

free groups. T o con clu d e a sim ilar resu lt for p o sitiv e c h a ra c te ris tic v a rie tie s , we must con sid er homotopy typ es com pleted away from the c h a ra c te r is tic of k . T he n o n -exactn ess of com pletion requires us to co n sid er e ta le neighbor­ hoods and only one prime at a time.

The proof we provide for Theorem 1 0 .7

is somewhat sim pler than the original one given in [30].

110

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

T H E O R E M 1 1 .7 .

Let X

c a lly c lo s e d fie ld k ,

let x

prim e in v ertib le in k . in X

be a smooth a lg e b ra ic variety over an a lg e b ra i­ b e a c lo s e d point of X , and let I be a

T h en th ere e x is t s an eta le n eigh bo rh o o d V

su ch that ttj(Ve j.)

of x

is a s u c c e s s i v e e x ten sio n of fin itely g e n e ra te d ,

fr e e pro-£ gro ups and s u c h that the natural map

V

^ K( ^ ( V e t / , l )

is a w eak e q u iv a le n c e . P roof.

The proof of Theorem 1 1 .6 provided by M. Artin in [7] proceeds by

verifying the e x is te n c e of a Z arisk i open Un of X elem entary fibration f n : Un -> therefore smooth). that R 1f ^ Z / £

Let

g:

is open in

(and

, where f ^ :

-» U ^_i

by g . If we ite ra te th is procedure (n ext co n sid e r­

ing the smooth sch em e V of x

P n_1

Un l be a fin ite, e ta le map ch osen so

=* g * R 1fn^ Z /£ is co n stan t on

is the pull-back of f

borhood

where

together with an

in p lace of X ), we obtain an e ta le neigh­

in X w hose stru ctu re map fa cto rs as a s u c c e s s io n of

elem entary fibrations

V = Vn - ^ V

with the property that R 1g -;+cZ /£ Let

-----------> Vj — —» Spec k

l —

is co n sta n t on V-_^

for e ach

i.

denote the geom etric fibre of g^. As argued in the proof of

Theorem 1 1 .5 , we may apply Theorem 1 0 .7 to conclude that (C^)et -> fib((g^)e j.) induces an isomorphism in

Z /2 cohom ology.

B ecau se

C-

is

con n ected and has £-cohom ological dimension 1 , the actio n of 771((V^_1) e ^) on H *((C j)e t ,Z /£ ) co n stan t).

“ H ^ fib C gP gj.Z /E ) is trivial (s in c e

is

We may therefore com pare the Serre s p e c tra l seq uen ce for (g j)e f-

f and (g j)et to con clu d e that fib ((g i) e t) phism in

I ^ g ^ Z /E

I fib ((g i) e t) induces an isom or­

Z /£ cohom ology.

We conclude that the com position (C i)et - f i b ( ( g i)e t) - f i b ( ( g i4 )

11. APPLICATIONS TO GEOMETRY

induces an isomorphism in len ce

(Cp)^.

Z /£ cohomology and therefore a weak eq uiva­

fib((g^)^t) of £-nilpotent pro-sim p licial s e ts . B e c a u s e I (C p ^ is weakly equivalent

is a smooth, co n n ected , affine cu rve over k , to

111

K (77,l) where

77 is a finitely gen erated , free pro-£ group. The

theorem now follow s im mediately by applying the long e x a c t homotopy seq u en ce to the s u c c e s s iv e fibering

12.

APPLICA T IO N S TO F IN IT E CHE V A L L E Y GROUPS

In this ch ap ter, we employ e ta le homotopy theory to determ ine cohomology groups of finite C hevalley groups and homotopy typ es of a s s o c ia te d K-theory s p a c e s . group of Fq-ration al points

In p articu lar, our techniques apply to the G(F^) of an alg eb raic group

over a field

k of finite c h a ra c te r is tic and to related tw isted groups (for exam ple, to

Un(Fq) = l(« ij) € G L n(F q ), ( a y ) , ( a j ) 1 = I„} ). The b a sic theorem of th is chapter is Theorem 1 2 .2 which provides an e ta le homotopy th e o re tic interpretation of an isomorphism of S. L an g. Although Theorem 1 2 .2 was first proved by the author in [34], the re le ­ van ce of L a n g ’s isomorphism w as first observed by D. Quillen in [62]; Quillen a ls o su gg ested formulating the L an g isomorphism a s a homotopy c a rte s ia n sq uare.

C orollary 1 2 .3 re la te s a sta b ility resu lt for the co h o ­

mology of finite c l a s s i c a l groups, w hereas C orollary 1 2 .4 provides a com parison of the cohom ology of d iscre te C hevalley groups to that of the cla ssify in g s p a c e s of the a s s o c ia te d L ie groups.

P rop osition 1 2 .5 d e te r­

mines certain (u n stab le) s p a c e s obtained using Q u illen ’s plus co n stru ctio n . The most in terestin g ap p licatio n of Theorem 1 2 .2 is given in Theorem 1 2 .7 , in which the various unitary K-theory s p a c e s for finite field s are identified. As the reader can a s c e rta in by comparing the m aterial of th is ch apter to the author’s various papers on th e se to p ic s , our p resen tation here is som e­ what sim pler and more d irect than that in the literatu re. The following proposition is a g en eralizatio n due to R . Steinberg [68] of a theorem of S. L an g [5 3 ] to tw isted C h ev a lley groups G ^(k)^ .

112

113

12. APPLICATIONS TO FIN ITE C H E V A L L EY GROUPS

P R O P O S I T I O N 1 2 .1 .

L e t Gk b e a c o n n e c te d lin ea r a lg e b ra ic group over

an a lg eb ra ica lly c lo s e d fie ld and let 0 : Gk -» Gk be a s u rje c tiv e endom orphism su ch that the group of k-rational points of Gk invariant under , H = G k(k ) ^ ,

is fin ite.

T h en the “ L a n g map” 1 / 0 : Gk -» Gk

is a p rincipal H -fibration, w here 1 / 0 g * 0 ( g ) 1 • C o n se q u e n tly , 1 / 0 l/

0

s e n d s a k-rational point g

to

in d u c e s an isom orphism : G k/ H ^ > G k . ■

T h is proposition is particularly striking when one re c a lls that top ologi­ c a l groups have no non-abelian con nected covering s p a c e s .

T hu s, P ro p o si­

tion 8 .8 im plies th at whenever H (a s in P rop osition 1 2 .1 ) is non-abelian and G red u ctiv e, the order of H must be d iv isib le by the residue c h a ra c ­ te r is tic of k . The following theorem (Theorem 2 .9 of [3 4 ]) is our e ta le homotopy th eoretic in terpretation of P rop osition 1 2 .1 .

The square ( 1 2 .2 .1 ) has been

referred to a s the “ co h o m o lo gica l L a n g fib re s q u a r e d 1 T H E O R E M 1 2 .2 .

L e t G(C)

be a com p lex red u c tiv e L ie group, let G ^

b e an a s s o c ia t e d C h ev a lle y in tegra l group s c h e m e , let k be an a lg e b ra i­ ca lly c lo s e d fie ld of c h a ra c te ris tic p , endom orphism with H = Gk(k )^

and let 0 : Gk -> Gk be a s u rje c tiv e

fin ite.

T h en a c h o ic e of em bedding of the

Witt v ecto rs into C d eterm in es a com m utative sq u a re in the homotopy ca tego ry for any prim e I with p | I : B H -----------------------------

( 1 2 .2 .1 )

(Z /O * , ° S in g.(B G (C ))

D

o S in g.(B G (C ))

(Z /Q ^ ° S in g .(B G (C )x 2 )

114

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

with the property that a c h o ic e of map on homotopy fib re s fib(D) -> fib(A) d eterm in ed by { 1 2 .2 .1 ) in d u c e s isom orphism s in Z /£ A

cohom ology, w here

is in d u ced by the diagonal G -» Gx2 .

P roof.

We interpret P rop osition 1 2 .1 as a sse rtin g that the following square

of sim p licial sch em es B (G £ 2 /A (G k),G £ 2 ,* )

B (G k/H ,G k,* )

( 12 .2 .2 )

d

S

1 X(f)

BGk

is ca rte s ia n , with the isomorphism on fibres by the L ang isomorphism

Gk/H

Gk/H

Gk2 /A (G k) given

Gk . The hypotheses of Theorem 1 0 .7

are sa tisfie d with M = Z / £ , b e ca u s e for e a ch sim p licial degree

n both

dn and Hi (SOn(F q) , Z / 0

i < n -2

H^(Spinn+1(F q ), Z /£ ) - .H ^ S p i n ^ F ^ Z / e )

i < n -2 . ■

Another ea sy co n seq u en ce of Theorem 1 2 .2 is our next co ro llary which show s that B G (F )

is a good cohom ological model for B G (C ). Corollary

116

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

1 2 .3 in clu d es P rop osition 1 .3 of [32] (all of the c l a s s i c a l ty p e s) and P rop osition 5 .3 of [52] (e a ch of the e x cep tio n al typ es with £ not dividing the order of the Weyl group). C O R O L L A R Y 1 2 .4 .

A s su m e the notation of T h eo rem 1 2 .2 and let F = F p ,

the a lg eb ra ic c lo s u re of Fp . A s s u m e that G is eith e r of c la s s ic a l type or that H * (T (C ),Z /£ ) -> H *(G (C ),Z /£ ) is s u rje c tiv e , w here T

is a maximal torus of G .

T h e d irect limit ( with

r e s p e c t to q = p^ ) of the maps Dq : BG (Fq) - (Z /Q ^ o Sin g.(B G (C )) a s s o c ia t e d to the fro b en iu s maps cf>^ : Gk -> Gk in d u ce s isom orphism s in hom ology a nd cohom ology H ^ B G C F X Z /Q P roof.

H * (B G (C ),Z /£ ), H *(B G (C ),Z /£ )

C learly , it su ffice s to con sid er homology.

from the p rojection dq : B (G k/G (F q ),G k,* )

H * (B G (F ),Z /£ ).

The map Dq a ris e s

B G k , so that it su ffice s to

show that th e se maps induce an isomorphism colim H *(B (G k/G (F q),G k ,* ) ,Z /£ )

H *(BG k,Z /£ ) .

Using the colim it of the Serre sp e c tra l se q u e n ce s for e ach (dq)e t , we conclude that it su ffice s to prove that colim H^(Gk/G (F q) ,Z /£ ) = 0 . To determ ine the map in homology induced by Gk/G (F q) -> Gk/G (F q A , we fit th is map and the L ang isom orphisms into the following comm utative square Gk / Q( F q ) ----------------------

i/^ ,q

i

I Gk

0 q'/q

c Gk

12. APPLICATIONS TO FIN IT E C H E V A L L EY GROUPS

where

117

0 q 7 q is the product of the maps cffl : Gk -> Gk for 0 < i < t with

q ' = q^. We readily verify that the re strictio n of 6 ^ ^

to T k induces

d * 7* : H j (T k,Z /£ ) -» H jC T ^ Z /£ ) given a s m ultiplication by 1 +q + ••• +q* 1 B ecau se

H *(T k,Z/f?) is generated by H 1( T ^ ,Z /£ ) ,

# ? '/ q - - K ( T k ’ ZM is the

we conclude that

- H / r k,z /o

O-map w henever £ divides ( q - l ) / ( q —1 ) = l + q + - " + q t _ 1 .

If H ^ T jZ /Q -> H ^ G ^ /E )

is s u rje ctiv e , the naturality of Prop osition

8 .8 im plies that ^ 7 q :H *(G k , Z / £ ) . H * ( G k,Z/l>) is a ls o the O-map for t su fficien tly large, so that colim H*(Gk/G k(F q) ,Z /Q = 0 as required.

The remaining c a s e s to co n sid er are

£ = 2 and q odd.

G = SOn o r.S p in n ,

In this c a s e ,

r f :H *(G k .Z / 2 ) _ H *(G k,Z / 2 )

is seen to be the id entity, b e ca u se

can be viewed a s the re strictio n

of : H ^ S O j Z ^ ) -> H ^ (S O ,Z /2 ), Q V f : H * (S p in ,Z /2 ) - H *(S p in ,Z /2 ) (a s in the proof of Theorem 1 2 .7 below).

T hu s,

tf * :H * (G k,Z / 2 ) -> H*(Gk,Z / 2 ) is the O-map for t even ( 0 ^ /^ *

is m ultiplication by t on prim itives in

H *(G k,Z / 2 ) ), so that colim H *(G k/G (F q); Z /2 ) - 0 . ■

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

118

If G is a reductive group over an a lg e b ra ically clo se d field of c h a ra c te r is tic G ^ (F)

p and if F = F p , then the commutator subgroup of G (F ) =

is p erfect.

We re ca ll th at the Quillen plus co n stru ctio n [43] B G (F ) -> B G (F )+

(with re sp e ct to the commutator subgroup) induces an isomorphism in homology and is the ab elian izatio n map on fundamental groups. Using a weight argument, one can e a sily verify that no cohomology c l a s s of H * (U (F ),Z /p ) G ^,

where

is invariant under T (F ) for any reductive group

is a maximal torus of G^ and

rad ical of a minimal p arab olic.

B ecau se

is the unipotent

U(Fq) con tain s a p-Sylow sub­

group of G(Fq) for any q = p ^ , we conclude that H * (G (F ),Z /p ) = 0 (c f. [64] or [34], P rop osition 4 .1 ). T his vanishing of the

Z /p-cohom ology of G (F ) and C orollary 1 2 .4

enable us to determ ine the homotopy type of B G (F )+ (a s in [3 2 ], Theorem 2 .2 ), P R O P O S I T I O N 1 2 .5 .

L e t G d en o te eith er G L n , S L n , SOn , Spinn or

Sp2n ior som e n > 0

and let F = Fp for som e prime p .

B G (F ) -> (Z /^ oo ° S in g.(B G (C ))

T h en the maps

determ in e a map ( unique up to homotopy

on fin ite sk eleta ) B G (F )+ -» w hich can be id en tifie d with the fib re

o Sin g.(B G (C )) Q /Z ^ ( B G ( C ) )

of the map

S i n g .( B G ) - .( Z (p)) ooo Sin g .(B G ), w here Z ^

is the su b rin g of Q c o n s is tin g of rationaIs w hose denom ina­

tors are not d iv is ib le by p . P ro o f. We employ D. S ullivan’s “ arithm etic fibre sq u a re ” technique to con clu d e that the vanishing of H * (B G (F ),Z /p )

and

that the maps B G (F ) -> (Z /Q ^ o S in g.(B G (C ))

H *(B G (F ),Q ) imply

12. APPLICATIONS TO FIN IT E C H E V A L L EY GROUPS

119

determ ine a map B G (F ) -> S in g .B G (C ). The uniqueness up to homotopy of th is map when re stricte d to finite sk e le ta of B G (F ) is given by ([1 3 ], VI. 8 .1 ).

B ecau se

B G (C ) is simply co n n ected , th is map fa cto rs uniquely

through a map B G (F )+ -> S in g .(B G (C )); b e ca u s e and

B G (F ) has triv ial

Z /p

Q homology, this la tter map uniquely fa cto rs through a map B G (F )+ ^ ( Q / Z (p))(B G (C )) . B y C orollary 1 2 .4 , th is map induces an isomorphism in integral

homology; b e ca u se

771(B G (F )+) is ab elian , we con clu d e th at this map

induces an isomorphism on fundamental groups.

For

G = S L n , Spinn , or

Sp2 n > B G (C ) is 2-co n n ected so that ( Q /Z ^ ) ( B G ( C ) )

is simply co n ­

n ected ; the Whitehead theorem then im plies that B G (F )+ -> ( Q / Z ^ ) ( B G ( C ) ) is a homotopy eq u iv alen ce.

B ecau se

772(S in g.(B S O n(C ))) - ^ ( ( Z ^ is su rje ctiv e for all

p,

o Sing.(BSO n(C )))

we sim ilarly con clu d e that BSOn( F ) ->

( Q /Z ^ p (BSOn(C )) is a homotopy eq u iv alen ce. F in a lly , B G L n( F ) + -* (Q /Z ^ ^ (B G L n(C ))

is a homotopy eq u iv alen ce, b e ca u s e the induced map

on universal covering s p a c e s is

B S L n( F ) + -» (Q /Z ^ p (B S L n( C ) ) . ■

In order to s ta te the most in terestin g con seq u en ce of Theorem 1 2 .2 , we must re ca ll the following definition (D efinition 1 .2 of [34]) of the c la s s i c a l C h ev a lle y gro u p s. D E F I N I T I O N 1 2 .6 .

Let

q

b e a p rim e p o w e r .

Then

F 'P q

d en o tes the

s i n g u l a r c o m p l e x of t h e h o m o to p y t h e o r e t i c f i b r e of t h e map

d ( l , ^ ) : B G ( C ) -> B G (C ) in the following four c a s e s , given with d isc re te group G(F^) (i) rep resen ts

G(Fq ) equals

G L (Fq) and d (l, *Fq L ) : B G L (C ) -> B G L (C )

on com plex K-theory.

120

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

(ii) se n ts

G (F ) eq uals l - 1?^

SO(Fq) and d ( l , ^ Q) : BSO (C) - B SO (C ) repre­

on oriented re a l K-theory.

(iii) G(Fq) equals

Sp(Fq) and d ( l , ^ s p ) : B Sp(C ) -* B Sp (C ) rep resen ts

on sym p lectic K-theory with q odd. (iv)

G(Fq ) equals

U(Fq) and d Q ,1? ^ ) : B G L (C ) -> B G L (C ) rep resen ts

l _ ^ _cl on com plex K-theory. ■ The following theorem , the fundamental resu lt of [34], e a sily d eter­ mines the K-^heories of fin ite fie ld s (the homotopy groups of B G L (F q)+ , B S O (F )+ , B S p (F )+ , and B U (F )+ ). T h e se “ u nitary” K-groups of the T. T. 4 finite field Fq have been tabulated in Theorem 1 .7 of [34]. The original com putation of ^ (B G L ^ F q )* )

w as ach ieved by D. Quillen in [64] by

other methods. T H E O R E M 1 2 .7 .

With the notation of D efin itio n 1 2 .7 , there a re homotopy

e q u iv a le n c e s a s s o c ia t e d to the sq u a res (1 2 .2 .1 ) X :B G (F q)+ - F 4 * for ea ch of the four c a s e s P roof.

G = GL , SO , S p , an d U

of D efin itio n 1 2 .1 .

One verifies th at H *(BG (Fq),Z /p ) = 0 where q is a power of the

prime p by using a vanishing range for H *(B G (Fqd ),Z /p ) as

which in c re a se s

d in cre a se s and by employing the transfer H *(BG (Fq) ,Z /p )

([3 4 ], Theorem 1 .4 ).

H *(B G (Fqd ),Z /p )

B e c a u s e both BG (Fq )+ and F 'P q are sim ple s p a c e s

(in fa c t, infinite loop s p a c e s ) with triv ial cohomology for c o e ffic ie n ts , it su ffice s to exhibit y

Z /p

and

Q

such that

X * :H * (B G (F q)+,Z /£ ) is an isomorphism for all primes £ ^ p .

H *(F «I* .Z /f ) As in the proof of P rop osition 1 2 .5 ,

we conclude that it su ffice s to exhibit maps

12. APPLICATIONS TO FIN ITE C H E V A L L EY GROUPS

121

: B G (Fq) - ( Z / O ^ F ^ for each prime £ ■/= p inducing isomorphisms X ^ H * ( ( Z / £ ) oooF ^

)Z /£ ) - H *(B G (Fq) ,Z /£ ) .

The colim it of ( 1 2 .2 .1 ) with re sp e ct to

n for

= Gn ^ equal to

G L n k , S 0 n k , Sp2n k , or G L n k and q determ ines a com m utative square BG (Fq)

( Z / £ ) „ o S ing.(B G (C ))

(1 2 .7 .1 )

(Z /£ l

( Z / £ ) „ ° S in g.(B G (C ))

o S in g.(B G (C )x 2 ) .

As is shown in P rop osition 2 .1 1 of [34], the lower horizontal arrow of (1 2 .7 .1 )

is

1 x

, sin c e

: BG n ^ -> BG n ^ and the homotopy equiva­

le n ce s of P rop osition 8 .8 determ ine maps ( Z / t ) " o S in g.(B G n(C )) - ( Z / f ) ^ o Sing.(B G n(C )) which sta b iliz e to ^

: B G (C ) -> BG (C ) (so that

( )t c ( )_1 o (^q : BGLn^k -> BGLn>k determ ines

: B G L (C ) -> B G L (C )) .

By definition of F 'P q ,

^ ts *n a homotopy c a rte s ia n square

F l^

Sing.(BG (C)>

( 1 2 . 7 .2 ) l x 'P j S in g.(B G (C ))

Sing.(BG (C)> x2

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

122

are simply co n n ected ,

(Z /l)^

B ecau se

B G (C ) and B G (C )x2

applied to

(1 2 .7 .2 )

yield s another homotopy c a rte s ia n square ( Z / l ) ^ ° (1 2 .7 .2 ) .

T herefore, we may ch o o se a map betw een sq uares (in the homotopy categ o ry ) 0 ( 1 2 .7 .1 ) - ( Z / E ) ^ o ( 1 2 .7 .2 ) with the map on upper left corner defined to be

and the map on the

other corn ers taken to be the identity. T he fa ct that X p H * ((Z /£ )oooF ' P ^ Z / 0 - H *(BG (Fq),Z /Q

is an isomorphism follows from the observation that £

induces an isomor­

phism of Eilenberg-M oore sp e ctra l seq u e n ce s (or th at £ re stricted to left v e rtica l arrows induces an isomorphism of Serre sp e ctra l se q u e n ce s). ■ An ap plication of Theorem 1 2 .7 proved by the author and S. Priddy is given in [41].

T he reader is referred to that paper for d e ta ils .

13.

FUNCTION C O M P LE X ES

R ecen t work by the author [39] and by the author and W. Dwyer [27] relatin g algeb raic K-theory to to p ological K-theory has required the use of function com p lexes of e ta le to p o lo g ical ty p e s.

T his chapter is intended

to provide the foundational m aterial n e ce s sa ry for th ose ap p licatio n s as well as th ose envisioned in the future. Much of th is ch apter (P ro p o sitio n s 1 3 .4 , 1 3 .6 , and C orollary 1 3 .7 ) is devoted to proving that maps of domain or range satisfy in g ce rtain proper­ tie s induce homotopy eq u iv alen ces of function com p lexes.

T h e se resu lts

enable one to use the fin iten ess theorem s of C hapter 7 and the com parison theorems of Chapter 8 to partially identify various function com p lexes. The s p e cific situ atio n relevant to alg eb raic K-theory is treated in P ro p o si­ tion 1 3 .1 0 , based on fin ite n e ss properties verified in C orollary 1 3 .9 .

The

relation sh ip of th e se function com p lexes to alg eb raic stru ctu res is d escrib ed in P rop osition 1 3 .2 . We re c a ll th at the function co m plex Hom(S. ,T .) sim p licial s e ts

S. and T.

(with b ase points

s

a s s o c ia te d to pointed

and t re sp e c tiv e ly ) is

the pointed sim p licial s e t w hose s e t of n-sim plices is the s e t of maps of pairs of sim p licial s e ts Homn(S. ,T ) = Hom((S. x A [ n ] ,{s } x A [n]), (T . ,t)) . F o r all pointed sim p licial s e ts

R . , S. , T . , there is a can o n ical

isomorphism Horn. (R . ,Hom. (S. ,T .) ) = Horn. (R. where R.

a

S. = R. x S. / ( i r ! x S. U R . x Is I).

then Horn. (S. ,T .)

a

If T .

S. ,T .) is a Kan com plex,

is a lso a Kan com plex so that the n-th homotopy

123

124

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

group of the component of Horn. (S. ,T .)

containing

of homotopy c la s s e s of maps F :S . x A[n] -> T .

f : S. -> T.

c o n s is ts

re la tiv e to

f ° p r 1 U t : S. x s k j ^ A[n] U { s ! x A[n] -» T . . We extend

Horn. (

, ) to a functor

Horn. ( , ) : p ro-(s. s e t s * ) x pro-(s. s e t s * ) -> p ro-(s. s e t s * ) by defining (1 3 .1 )

Horn. (iS? ; i e l!, ! t ! ; j eJ j) = icolim {Horn. (S! , T J! ); i elS; j eJ 5 ■ I

In the following proposition, we introduce the alg eb raic function com ­ plex and re la te it to the function com plex of e ta le to p o lo g ical ty p es. P R O P O S I T I O N 1 3 .2 .

For p o in ted sim p licia l s c h e m e s

X. ,x

and Y . , y ,

we d efin e the a lg e b ra ic fu n ctio n com plex Horn. (X . ,Y .) e ( s . s e t s * ) ( with b a s e points im plicit) to b e the sim p licia l s e t w hose n -sim p lices a re maps of pairs (X . ® A [n], ix \® A[n]) -> (Y . ,y ) .

If X .

and Y .

a re locally

no etherian, then th ere is a natural map of pointed sim p licia l s e ts Horn. (X . ,Y .) -

lim

Horn. ( ( X .) e t , ( Y .)e t ) •

H R R (Y .)

P ro o f. T his map is obtained by observing that an n-sim plex of Horn. (X . ,Y .)

represented by X . ® A[n] -> Y .

determ ines

(X . ® A [n])et ->

( Y .) et in lim Hom0((X . ® A [n])e t ,( Y .)e t ) and thus in

lim Hom0(( X .) e t x A [n ]x A [n ]/S x ix A [n ],(Y .)e t )

«

lim Homn( ( X .) e t ,(Y .)e t)

by Prop osition 4 .7 . ■ As an immediate corollary of P rop osition 1 3 .2 , we con clu d e the e x is te n c e of a natural map

13. FUNCTION COM PLEXES

(1 3 .3 )

Horn. (X . ,Y .) -

holim

125

Horn. (( X .)e t ,( Y .) e t)

H R R (Y .)

thanks to the natural transform ation B ecau se

lim( ) -> holim( ) of [12], X I. 3 .5 . W. of pointed Kan com p lexes induces a pointed homotopy betw een the maps f* ,g * : Horn, ( { s !i ,T .) -> Horn. ({s !},W .) (of pointed Kan co m p lexes).

C onsequently, we im mediately conclude the

following. P r o p o s i t i o n

1 3 .4 .

L e t {SM

p ro-(s. s e t s ^ ) .

€

T h en Horn. (Is il,

)

d eterm in es a functor Horn. (IS !!,

) : pro-H^ -> pro-K^ .

In particular, if 1t!S -» {w!*} is a map in pro^Kan^) ( w here (Kan^) the fu ll su b ca teg o ry of (s . s e ts ^ ) an isom orphism in pro-K^ ,

is

co n s is tin g of Kan c o m p le x e s ) w hich is

then

Horn.

- Horn. ({SM, iW^i)

is a lso a map in pro-(KanH 0,

there is a natural isomorphism

Hom-r/ ( 2 mS. , T . ) ,

77m(Hom. (S. ,T .))

where the homotopy groups of Horn. (S. ,T .) are th ose

based a t the point map S. -> T . U A[m] x I s !) for m > 0 and pro-(s. s e ts ^ ) and T . 77m(Hom. ( l S !!,T .))

a pointed Kan com plex.

^

and where

S mS. = A[m] x S ./( s k m_ 1 A [m ]x S .

£ ° S . = S. . More gen erally, for Is ! ;i< rl! 0 .

C onseq u ently, for any {SM e p ro-(s. s e ts ^ ) and any

iT? ; je jS