Fibrations in etale homotopy theory

FIBRATIONS IN ETALE HOMOTOPY THEORY

by ERIC M. F R I E D L A N D E R

Let f : X - + Y be a map of geometrically pointed, locally noetherian schemes and let A be a locally constant, abelian sheaf on X. We derive the Leray spectral sequence

E2V'~ = HP(Y, Rqf.A) =-HV+q(X, A)

as a direct limit of spectral sequences obtained from the simplicial pairs constituting f0t, the etale homotopy type o f f . This " simplicial Leray spectral sequence " can be naturally compared to the cohomological Serre spectral sequence for let and A. We employ this comparison to investigate the map in cohomology induced by the canonical

map (Xv)0t-+I~(f/t), where b(f/t) is the homotopy theoretic fibre o f f 0 t and X v is the geometric fibre o f f .

Of particular interest are the special cases: I) f : X--->Y is proper and smooth, and 2) f : X - + Y is the structure map of an algebraic vector bundle minus its zero section. In these cases,

H*(b(Z,), A)-~H'((X~)0,, A)

for any locally constant, abelian sheafA on X with finite fibre whose order is not divisible by the residue characteristics of Y. With certain hypotheses on the fundamental groups involved, wc obtain a long exact sequcncc of homotopy pro-groups:

. . . f - - -->re X -+7: Y -+~z 1 ~-- - + . - - -~.((X~)o,) n(^ 0,) o(^0,) n- ((X~)00

where ( ) denotes profinite completion " away from the residue characteristics of Y "

We conclude with a proof of Adams' Conjecture concerning the kernel of the J-homomorphism on complex vector bundles over a finite C-W complex [i]. The proof is a completion of a proof sketched by D. Quillen [i3] , employing the result

that an algebraic vector bundle minus its zero section has completed etale homotopy

type equal to a completed sphere fibration.

We caution the reader to remember that throughout this paper, a sheaf F on a

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scheme X will be a sheaf on the etale site of X; thus, the cohomology on X with coefficients in F, H'(X, F), will be etale cohomology [3].

The author is deeply indebted to M. Artin for his continued interest and guidance. Furthermore, section 3 in its present form (in particular, Theorem (3.9)) is due to the referee, whose suggestions have proved most valuable.

x. The Category J / A s s o c i a t e d to a Map o f Schemes

After establishing some notation, we recall the definition of a special map of simplicial schemes and of an etale hypercovering of a scheme. Relativizing the category of etale hypercoverings of a scheme, we introduce the category Jt associated to a map f : X-+Y of schemes. This category Jt is seen to be codirected with cofiltering homotopy category, Ho Jr" In subsequent sections, we shall study various functors defined on Jt and Ho Jr; in particular, the etale homotopy type o f f is a pro-object indexed by Ho Jr"

Let a scheme X be given. Denote by (Et/X) the category of schemes etale over X, with maps in (Et/X) covering the identity map of X. Let A~ denote the category of simplicial objects U of (Et/X). I f X is a " pointed scheme " with given geometric point x : Spec ~x-+X, let A~ denote the category whose objects U. each have a chosen geometric point u : Spec ~z---~(U.)0=U0 above x, and whose maps U'.-+U. are pointed, simplicial maps.

Given a (pointed) scheme X and a (pointed) simplicial set S , let S. in A~ denote the naturally associated (pointed) simplicial scheme: (g),~ is given as the disjoint sum of copies of X indexed by the set S,, and the face and degeneracy maps of g are induced by the face and degeneracy maps of S . In particular, A[o] denotes the final object of A~ Given two maps f , g : U~=~U. in A~ a categorical homotopy connec t ing f and g is a map U:xXA[I]-+U" in A~ restricting t o f and g (and

furthermore, sending (SOU" , oi) to SoU if X is pointed). - - - - t A map U~-+U. in A~ is special provided that U 0 ~ U 0 is surjective and

provided that for each n>o

U.-+(cosk._lU'.). • U.

is surjective (recall that the functor cosk,( ) is right adjoint to the truncation functor). One can readily check that the composition of special maps is special; moreover, the pull-back of a special map by any simplicial map is special. An object U. of A~ is called an etale hypercovering of X provided that the unique map U ~A[o] is special.

We denote by Jx the category whose objects are etale hypercoverings of X and whose maps are special maps in A~ We denote by Ho Jx the homotopy category ofJx: a map in Ho Jx is an equivalence class of maps in Jx, where the equivalence relation is generated by pairs of maps connected by a categorical homotopy.

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Definition (x.z). - - Let f : X-~Y be a (pointed) map of schemes. Define Jf to be the following category: the objects of Jr are pairs U.-+V., with V. inJy , U. inJx , and the induced simplicial map U . - + V x X in Jx (i.e., special); a map

Y

u'. u.

V: V.

in Jf is a pair of maps, V:--~V. in Jy and U:-+U._x V: in Jx" v. i,,,

A connected category J is said to be codirected provided that any diagram ./~k

in J can be completed to a commutative square 7i 'x "/ I,,~ .,,,, k. ]

A codirected category J is said to be cofiltering provided that for any pair of maps jz~.k in j , there exists a map i ~ j i n J such that the compositions i-+j=~k are equal.

We proceed to relativize Verdier's proof that Jx is codirected and Ho Jx is cofiltering ([3], V, App.).

Proposition (x.2). - - Let f : X-+Y be a (pointed) map of schemes. Then Jf is codirected.

Proof. - - J t i s connected : for given U'-+V' and U ~ V in Jr, then U ' x U -+V'xV. . . . . . X " Y

is in Jr and the projection maps are maps in Jr" Let the diagram

in Jr be given. Then U'.' x U. -+ V" X V. is in Jr and the projection maps are maps u'. v ' ,

in Jr, as can be immediately checked using the following sublemma. Sublemma. - - I f

U' . ' ~W" and U.-+U'._xW. are special, w'.

is a commutative diagram in A~ such that then U'o' x U~ -+ W " X_ W is special.

u: w"

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Proof of sublemma is immediate upon observing that the maps

U'.' x U. -+ U'.' x (U'._x W.) and U'.' _x (U'._x W.) = U:'_x W. -+ W'.'_x W. c: lJ: w: u'. w: w: w:

are special.

We denote by Ho Jt the " homotopy category " of Jr: the objects of Ho Jt are those of Jr; a map in Ho Jr is an equivalence class of maps in,It, where the equivalence relation is generated by doubly commutative squares

U'.

v'. v.

connected by a " categorical homotopy of pairs ": namely, a commutative diagram m

U : xX A [ I ] > U~

, V.

- - m _ _

such that U:XxA[I ] --> U. is a categorical homotopy in A~ V:~A[I] -+V. is a

categorical homotopy in A~ and U'.XxA[I ] ---> V'.~A[I] is induced by U~-->V'..

Employing the techniques Verdier used to prove that HoJx is coflltering, we shall verify that the " relative homotopy category " HoJt is likewise coffltering.

Given U'. and U. in A~ define the contravariant set-valued functor Hom.(U'., U.) on A~ by

Hom.(U:, U.)(U'.') = Hom(U'.' X x U'., U.)

for any U'.' in A~ If Hom.(U'., U.) is representable as a simplicial scheme in A~ (also denoted by Hom.(U:, U.)), then

Hom(U'.', Hom.(U'., U.)) ----- Hom(U'.' ~ U:, U.),

the usual adjoint relation between Hom( , ) and categorical products. Observe that Homo(A[n], U . )= U, and Hom0(sk~_tA[n], U.)=(cosk,_tU.) . for any U. in A~ and integer n:>o.

If S. is a simplicial set with finitely many non-degenerate simplices, Hom.(S., U.) is representable for any U. in A~ ; for each n_>_ o, Hom,(S., U.) =Hom0(h[n]XxS" , U.)

is a finite projective limit of components of U. occurring in dimensions < n+m, where m is the maximum of the dimensions of the non-degenerate simplices of S. In particular,

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categorical homotopies into U. are represented by the simplicial scheme Hom.(A[i], U ) . If U. in A~ is pointed by u, then Hom.(A[i], U.) is pointed by

sou : Spec ~ , -+ i]l----[-Iom0()X[I], U.)

and represents pointed categorical homotopies into U. The key step in Verdier's proof that Ho Jx is cofiltering is the verification that

Hom.(A[I], U'.) -+U'. • . is special whenever U: is a hypercovering of X. The following

lemma relativizes this result (which the lemma implies by setting U equal to A[o]).

Lemma (x. 3). - - Let X be a (pointed) scheme and let U:-+U. be a special map in

A~ Then the induced map

Hom.(A[i], U~) -+ (U:• • Hom.(A[I], U.) X (U. x U.)

X

is special in A~ Proof. - - In dimension o, Homo(A[I ], U~)=U~ maps surjectively onto

(UoXUo) x Homo(A[I],U.)~(UoXUo) X U 1 x (Uo x Uo) X (uo x Uo)

x x

since U'.-+U. is special. In dimension k;>o, we use the notation " ckkU" " and " H.(U~ to denote

(coskk_lU--.) k and Hom.(A[i], U.) respectively, for any If. in A~ We must prove that Homk(A[I], U'.)=Homo(A[k ] • U~) maps onto the following projective limit:

ck,H.(U:) x _ _ (U~xU~) • H,(U.). Ckk(U' • U.') • ekkn,(u.) X CUk ~Uk)

X ckk(U, xx U.)

We shall denote this projective limit by Pk. Let M o denote the simplicial set defined as the fibre sum

(skk-xA[k]xA[I]) (,kk_xA[k]I-[ SkoA[1])(A[k]xsk0A[x])'x

M o -+ skk(A[k ] xA[I]) -+ A[k] x A [ I ] c a n be factored as

~V~ 0---~" , . ~ Mk=Skk(A[k] xA[I ] ) ---~o , ~ ~ U2k.F l=A[k] xA[I]o

For o < i < k , Mi+ 1 is obtained from M~ by adjoining a k-simplex (so that

sk,,_l [k] , ,', [k]

is co-cartesian) ; and for k < i <2k, Mi+~ is obtained from M~. by adjoining a (k+ i)-simplex.

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For any U. in A~ Homo(Mo, U.)=ckkH.(U.) x (Uk• Okk(g.~u,) X

Pk = Homo(Mo, U:) • Hom0(A[k ] xA[I], U,). Homo(Mo, U.)

Therefore,

To verify that Hom0(A[k ] xA[I] , U:) --)- Pk is surjective, it suffices to check for each i, o< i < 2k , that

Homo(Mi+l, U~) -+ Homo(Mi, U~) X Homo(M~+t, U.) Hom0(Mi, U.)

is surjective. This follows directly from the hypothesis that U : ~ U is special and from the fact that M~.+t is obtained from Mi by adjoining a simplex to its skeleton.

The following proposition verifies that Ho Jr is a good relativization of Ho Jx.

Proposition (x.4). - - Let f : X ~ Y be a (pointed) map of schemes. Then HoJt is cofiltering and the source and range functors, s : H o J t ~ H o J x and r : HoJf -+HoJy , are cofinal.

Proof. - - Provided that Ho Jr if shown to be cofihering, cofinality of the source and range functors is easily checked. One simply observes that if U:--->s(U.-+V.) is a map in Ho Jx then there is a natural lifting

u: v.

V V

to a map in HoJr ; similarly, if V:--*r(U.--*V.) a natural lifting

u.xv: u.

v: 1 v: v.

To prove that Ho Jt if cofiltering, let

u: v.

v: v.

is a map in HoJy , then there is

be two given maps in Ho Jt" Define

U" H o m ( A [ I ] , U ) x _ U ' =

V:' Hom.(A[i], V.)_ • V: V. ~u

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Since the compositions m _ _ m

u7 u: u.

V:' V; V.

m m

are clearly equal in HoJ t , it suffices to prove that U " ~ V ' . ' is an object of HoJ t and that the projection map

v'.' v'.

v7 v:

is a map of Ho Jr" By Lemma (I.3) , H o m ( A [ I ] , V ) - - - ~ V ffV. is special; therefore, V'.'-+V~ is

special and consequently V'o' is an etale hypercovering of Y. One readily checks that Hom.(A [I], V. x X ) = Hom.(A [I], V.) • X . y Hence, U"-*V". . xXy factors as the compo-

sition of the two maps

Hom.(A [i], U.) _ x_ U: �9 Hom.(A [i], W X X) x U. x U _ x U: U . • Y ( ~ ' o X V . ) x X y Y X U . ~ U .

Hom.(A[i], V. xX) x U: > Hom.(A[i], V. xX) X V::xX.

Using Lemma ( i .3) , we conclude that

In order to prove that

U" -+ V" X X is special. Y

u'.' u:

v'.' v~

is in Ho Jr, it suffices to prove that U" -+ U' _x V" is special. This is immediate from Lemma (1.3). V:

We remark that the range functor r : J t ~ J y is fibrant, with fibre above V denoted by Jt, v. ([8], V I . 6 . 1 ) . Let HoJt, vr denote the homotopy category of Jt, v. whose homotopy relation is generated by pairs of maps in Jt, v. related by a categorical homotopy

u : x a [ q - . u.

V.xa[i] ~ V.

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With notat ion as above, the two compositions

U:'• V. U: U. V:'

V. V. V.

are equal in HoJf, v., where g, g' are objects in Jf, v. and where V ~V' . ' is the canonical map. Thus, HoJf ' v. is cofihering.

Given a scheme Y, let ET/Y denote the category whose objects are pointed, connected galois covers h' : Y ' ~ Y (h' is finite, etale and Y ' • and whose

Y maps h " ~ h ' are commutat ive triangles. Since there exists at most one map h"->h' between objects of ET/Y, ET/Y is cofiltering. Recall that the Grothendieck fundamental group =I(Y) of Y is the pro-group {Gal(Y'/Y)}ET~; thus, the cokernel of r:I(Y')-->=I(Y ) is Gal(Y'/Y), for Y ' ~ Y in ET/Y. (For details, see [8].)

Given a pointed map f : X-+Y of schemes, we define the category J?~ as follows. An object {' : U ' ~ V ' of Jy is an object of Jr• for some h' : Y ' -+Y in ET/Y. A map g" -+g ' in J~" is a map g"--->~' ~ Y " in Jt • The homotopy category of JT is

denoted H o J y : a map g"--->~' in HoJ~" is a map ~ " - + ~ ' • in HoJf• y ,

Corollary ( I .5 ) . - - Let f : X - + Y be a pointed map of schemes. As defined above, J[ is codirected and HoJy is cofiltering. Furthermore, J[ and HoJ~" are fibre categories over ET/Y, with respective fibres Jf• and HoJf • over h' in ET/Y.

Proof. - - Products and fibre products exist in J [ because they exist in ET/Y and in each Jt• The existence of coequalizers in each HoJt• implies the existence of coequalizers in H o J [ . By definition of JT and H o J [ , the inverse image functor exists: sending Y"-+Y' in ET/Y to the functor ~ ' t -*~ ' •

y ,

We conclude this section by remarking that if a functor ~ : F--*G is fibrant, then l i m = l i m . l i m on functors F~

F c Fg

2. F i b r e s o f ~ t .

Having recalled the " extended homotopy categories " of simplicial sets, gU 0 and :U0,~ir~ ([4], w I), we define the etale homotopy type of a map of locally noetherian schemes, let : H o J f ~ 0 , ~ i r s . We introduce canonical maps (X~)st~I)(f,t)--->I)(f,'t) between the " geometric, " " naive, " and " homotopy-theoretic " fibres off~ t. We then identify the map on cohomology induced by (X~)et-+ I)(f0t). In subsequent sections, we shall study the map t)(f , t)-+ [ ( f / t ) .

Given a locally noetherian scheme U, we denote by ~(U) the set of connected components of U. I f U. in A~ is a simplicial scheme over a locally noetherian scheme X, then we denote by =(U.) the simplicial set given by ~(U ),----n(U,). Clearly,

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m

the association ofr.(U.) to U. is functorial on A~ and sends categorical homotopies to simplicial homotopies.

We recall the following " extended homotopy categories " ;U 0 and 3U0,p~ir,, which are geometrically realized as the homotopy category of pointed C-W complexes and the homotopy category of pointed pairs of C-W complexes. Let Ex~~ ) denote the left adjoint to the inclusion functor of the full subcategory of Kan complexes in the homotopy category of simplicial sets [io]. Then ~0 denotes the category whose objects are pointed simplicial sets S., and whose maps from S. to T. are homotopy equivalence classes of pointed simplicial maps S.-+Ex~176 The objects of ~0,~ir, are pointed simplicial maps S.-+ T.; the maps of #U0, r*i,~ are equivalence classes of pointed commutat ive squares

S: ~ S.

E x i T '. > Ex |

where the equivalence relation is generated by doubly commutat ive squares connected by a pointed homotopy of pairs.

Definition (2 . I ) . - - The etale homotopy type of a locally noetherian, pointed scheme X is the pro-object in pro-~Cd o

X . t : H o J x -+ ~Y" o

induced by the connected component functor. The etale homotopy type of a pointed map f : X - + Y of locally noetherian schemes

is the pro-object in pro-~0,p~i, ~

fot : H ~ ~0.#r ,

induced by the connected component functor.

We remark that Proposition (1.4) implics that let detcrmincs a map Xet--+Yot in pro-X" o.

Given a simplicial map f : S.-+T. commutat ive triangle

s.

such that the inclusion S --+S: a Kan fibration ([7], VI) .

of simplicial sets, we functorially associate a

�9 s:

is a weak homotopy equivalence and such that f r is Furthermore, given a categorical homotopy of pairs

S:XtX[I] , S.

T : X A [ I ] , T.

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there is induced a categorical homotopy of triangles

S:X a[ I ] ~P- S .~,,,~

~ (st)'xa[q > s:

/ . T / T ; x A [ I ] "-

This " fibre resolution functor " therefore induces functors ~o,p~i. ~ ~0,p,i. and

pro-~0, p~i. ~ pro-.~0, p,i,,. Given a pointed simplicial map f : S.-+T., let I ) ( f ) = S . ~ t be the pointed

fibre over the distinguished point t of T. This definition determines a functor

I~ : pro-~0, ~ir, ~ pro-~0 .

The following proposition introduces the various fibres associated to f~t.

Proposition (2 .2) . ~ Given a pointed map f : X ~ Y of locally noetherian schemes, with

geometric fibre i : X . v ~ X . Then there exist canonical maps in pro-~g'o

from the " geometric fbre " to the " naive fibre " to the " homotopy fibre " of for. Furthermore,

these maps fit in a commutative diagram in pro-uY'o :

(x~) . , �9 I~(f.,) > ~(/:,)

X,t > X;~

Proof. - - The map D(fot)-~l~(f/t) is induced by the natural map in prO-~o, pair,, ___>. �9 fo, fg,.

To define (Xu),t-+I)(fot) in pro-.ff~0, we associate to each object U.-+V. in HoJ l an etale hypercovering of Xv, which we denote by 9(U.--~V.), and a map ~(,(u. -~ v.)) -~ ~(~(u. -+ v.)).

Given U.-+V. in HoJf, we let F denote the trivial simplicial sub-object of V.

generated by the distinguished component v of V o. We observe that U. x F -+ v • X = u u Y

is special and that b(~(U. -+ V.)) = ~(U. • V). We define ~(U.--+V.) to be U. • V • X v ~7. ~. u

and define w(q~(U.-+V.)) -+ D(~(U.-+V.)) to be the map induced by the simpliclal

map U.V.•215

I f u: u.

v: v.

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is a map in Jr, then

~(u: -+v ' . )= w:_x ~' • x~ -+ u._x ~ x v' x x ~ = u . x ~ x X ~ = ~ ( u . - ~ v . ) V-' u' V. ~ u' V. u

is a map in Jx, (i.e., special) and

= ( r ( u ' . ~ v' . ) ) ~ tg(,~(u'. ~ v' . ) )

~(~(u. ~ v.)) ~ I)(~(u. ~ v.))

commutes. The maps b(f~t) -+X.t and D(f/t) --*X~t are given as

I](At)-+soAt~-Xot and I](f:t)-+so/:t~-X:t (where s :~o,p~, . -* ~o functor). We must check the commutat ivi ty of the square

(xp. , , b(A,)

t

Thus the pairs n ( ? ( U - + V ) ) -+ l ) (n(U -+V.)) define a map (Xu)et -+I)(f~t). the compositions is the " source "

The composition (Xu).t -+ X,t ~ sol . t in li__m l im [n(W.), s ( n ( U . ~ V . ) ) ] is given by ao as ao Jxy

pairs ~(U xXXu) -+ n ( U ) ; whereas the composition (Xv).t -+ b(fot) -+ s~ is given by

pairs n ( U x ~ X Xu) -+ ~(U.). V. u

To check that these pairs determine the same map (Xv)ot-~ so for, we observe that U . x ~ x X u - * U. factors through the simplicial (though not special) map

V, U

U. x ~ x X u -+ U x X u of etale hypercoverings of X u. Let W be the product in Jx, ~. u " X

of U . x ~ x X u and U x X u. As in Proposition (I .4) , we find W ' - + W . inJxy with ~. u "X

W:-+W.z~U. x X X u categorically homotopic. Hence, the pairs =(U. x x Xu) -> ~(U.) and

~(U. x ~ ~ Xu) -+ r~(U.) induce homotopic maps n(W:) -+ n(U.). V.

In order to verify that Up becomes " arbitrarily fine " for U in Ho Jx, we verify the following lemma (asserted in [3], V, App.).

Lemma (2.3). - - For any scheme X and any integer p > o , there exists a functor R~( ) : (Et/X) -* A~ satisfying:

a) Rr ( ) sends surjective maps to special maps. b) R{() is right adjoint to the functor sending U. to Up. c) I f U. /s A~ Z ~ U p # a map in (Et/X), and U ' = U • R.P(Z), then

p r . : U~ ~ U p factors through Z-+Up. " ""P'(vP)

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Proof. - - Define t(p, m) to be the set of non-decreasing maps from {o, ..., p} to {o, ..., m} for any m > o . The t(p, m) naturally determine a co-simplicial set, denoted V(P), with V(p)m=t(p, m). We define R;(Z) to be the composition of V(P) :A~ (Finite sets) ~ with the functor n z : (Finite sets)~ (Et/X) given by sending a set S to the fibre product over X of Z with itself $(S)-times.

For any m > i, (cosk,,_lR.v(Z)),~ is the fibre product over X of copics of Z indexed by " deg"(V(p)) , " those co-simplices in the image of V(p) " -1 under some co-face map. Hence, R v(Z)m-+ (cosk,,_lR:(Z)), , is surjective, arising from an injective map of indexing sets deg'(v(p))-+V(p) ". We may thus readily check that

- + • 1 4,(z) (eoskrn_X Rf(Z))m

is surjcctivc whenever Z'-+Z is surjective and re>o; thus, a) is valid. Namely, let any geometric point

x • : Spec ~ -+ (cosk~_lR,v(Z')),,, • R~(Z) u (~oBk~n_ I R.P(ZJJm

be given; wc lift x• to xXw :Spcc ~-+ R~(Z') by defining w to bc some lifting of Y

those factors of Z : Spec ~ -+ P~(Z) not determined by y : Spec ~ -+ (cosk~_iP~.P(Z)),~. To check b), let f : Up-+Z in (Et/X) be given. Define U.-+R.P(Z) by sending U k

into the factor of (RP(Z))~ corresponding to ~o... 0qo ~m... ~h : A[p] -+ A[p--t] -+ A[k] via the composition fosho.., osho~ o... o~ : Uk-+U~_~-+Up-+Z. One may readily check that this establishes a i--I correspondence between Horn(Up, Z) and Hom(U., R.P(Z)).

t -f To check c), observe that the projection map pr1:Up~U p factors through any factor Up_x Z of U~. Yet the factor corresponding to the identity map in V Ip)

up cqu~s Z, since Up into the factor of R~(Up) corresponding to the identity map is the

identity.

The following proposition explicates the map on abelian cohomology induced by (Xu)0t-+~)(f0t). A similar statement can be made for non-abelian finite groups G and integers p = o , i .

Let X be a scheme and let F be an abelian sheaf on the etale site of X. We recall the Verdier isomorphism

lim HP(U., F) --% Hq(X, F) ao~

where H*(X, F) is the etale cohomology of X with values in F, where U. runs over etale hypercoverings in H o J x , and where H ' (U. , F) is the cohomology of the cochain complex F ( U ) ([3], V, App. ; or see Proposition (3-7) below). In particular, let A be a locally constant, abelian sheaf on the etale site of a locally noetherian scheme X, with corresponding local coefficient system on X,t also denoted by A ([4], w Io). Then H'(X0t, A ) = lim H~ A ) = lim H*(U., A) -~ H*(X, A).

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Proposition ( 2 . 4 ) . - - Let f : X ~ Y be a pointed map of locally noetherian schemes, with geometric fibre i : Xv-+ X . Let A be a locally constant, abelian sheaf on X . Then for each p > o, there exists an isomorphism

HP(~(f,t), A) --% (R2f.A)u

which fits in the commutative square

HP([~(f,t),A) --% (RPf.A)y

H ' ( (N) . t , i'A) --% H'(Xv, i'A)

where the left vertical arrow is induced by (X~)~t ~ I~(f~t), the right vertical arrow is the canonical map, and the bottom arrow is the Verdier isomorphism.

Proof. - - We recall that if U is an etale hypercovering of a scheme Z, if F is a sheaf on the etale site of Z, and if F ~ I " is an injective resolution of sheaves, then the spectral sequence of the bicomplex I'(U.) can be written

F_~,,=H'(U., Wq(F)) :~ Hs(Z, F),

since U is acycIic at each geometric point of Z. The association of

v . )=u .x xX, V. u

in HoJx " to U . ~ V . in HoJt induces a map of spectral scquences:

lira H (U.x V, li__m H (v• F) ao-~/ v. Ito sr Y

'E~, q= lira H~(W., ~fq(i*F)) =.- HS(Xu, i'F) Ho JXy

The map on abutments is precisely the natural map (RSf.F)v-~ HS(Xu, i'F). Taking F equal to A, we obtain E~'~ A) and 'E~'~ A); moreover F_~'~ '~ is induced by (Xv)0t~I~(f~t). The Verdier isomorphism is the edge homomorphism of the 'E p'q spectral sequence. If we define H~(I~(f.t), A) -+ (RPf.A)y to be the edge homomorphism of the E p' q spectral sequence, the square

Hp(D(f.t), A)

HP((Xu),t, i'A)

(RPf.A)v

1 , H'(X~, i'A)

commutes, for any p > o.

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18 E R I C M. F R I E D L A N D E R .

To show that HP([~(f~t), A ) ~ (K~f.A)v is an isomorphism, it suffices to verify that E~ 'q=o for q > o . Let q > o be given and let ?- be a cohomology class in HP(U._>< ~, 9r represented by a cocycle c in 9~(A)(U._• ~)p. Let c' in .~q(A)(U~,)

V. V. be the cochain obtained from c by extension by zero. Since ~ ( A ) vanishes at every geometric point, there exists an etale surjective map Z---~?~_Y r such that c' goes to o in 9/*q(A)(Z). By Lemma (2.3), the special map U ' = R P ( Z ) • U ~ U satisfies the

�9 " RSVp) " "

condition that U~--~Up factors through Z ~ U p . Hence, ? in Hq(U._x ~-, ~r goes to o in H~(U'._X~,o~(A)). Thus E~ 'q~o for q>o. v.

V.

3" The SimpHclal Leray Spectral Sequence.

In this section, we study a naturally constructed spectral sequence EP'q(g, A) associated to a map g : U ~ V of simplicial sets and to a local system A on U . After checking functoriality of this construction and after verifying that E p' ~(g, A) is the usual Serre spectral sequence whenever g is a Kan fibration, we proceed to identify E p' ~(fet, A) for f : X-+Y a (possibly pointed) map of locally noetherian schemes and A a locally constant, abelian sheaf on X. Theorem (3-9) asserts that EP'q(f,t, A) takes the form of the Leray spectral sequence. This " simplicial Leray spectral sequence " is by construction readily compared to the Serre spectral sequence for f , EP'q(j~rot , A).

For notational convenience, we shall denote sk~_xA[p ] by sk(p).

Construction (3. x ). - - Let g : U.--~V. be a map of simplicial sets and let A be a contravariant local system on U. . We construct a spectral sequence

E~' q(g, A)----HP+~(g-XVip), g-~Vip_~); A) =~ H~+~(U., A)

where V(~) denotes skpV. and where g-XVip ) denotes V(~,)• V

Namely, let A(U.) be the complex of cochains on U. with values in A. Define a decreasing filtration on A(U.) by FPA(U.)=ker(A(U.)-+A(g-XV(p))) . We obtain a

�9 "~.p,q__~'[2p, q + l spectral sequence (Eg,~, do)with and d0:-0 - - 0 �9 . lzp, q ~ p , q+x is the differential the differential o f P ' - ~ A ( U )/FJ'A(U ). Equivalently, do :_ o --~o

of ker(A(g-~V~pl) ~ A(g-~Vip_~))). Taking cohomology, E p q--HP+qt--tV "-~V �9 A) and dl : ~ P ' q ~ + ~ ' ~ is the 1 - - \ ~ (p), 6 ( p - - l ) , ~ l ~ 1

connecting homomorphism in cohomology for the short exact sequence

o-*A(g-~V(p + 1), g XV(p)) --~A(g- 1V(~ v~), g - lV(~-l)) --~A(g-xV(~), g- tV(p_ ~)) -~o.

The following lemma explicates the functoriality of E ~' q(g, A) as constructed above.

Lemma (3.2). - - Let U: U.

r Q' I' v : v .

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FIBRATIONS IN ETALE HOMOTOPY THEORY I9

be a map of simplicial pairs, let A be a (contravariant) local system on U , and let

be a map of local systems on U ' . Let

G*A-~A' Then G induces a map E p' e(G) : E p' q(g, A) -+ E p' q(g', A').

U:XA[I] ' > Uo

l Cx' l' V:xA[q 9% v.

be given together with a map h'A---~pr~A ' maps EP'q(g, A) =~ E p' q(g', A') are equal, beginning with the E1 term.

of local systems on U : x A [ I ] . Then the induced Furthermore, let

u:xA[x] ~, u.

v'.xh[~] ~, V.

be given together with a map h 'A~pr~A' of local systems on U:!xA[x]. !~*. q(g, A) ~ E p' q(g', A') are equal, beginning with the E 2 term.

Proof. - - To verify the first assertion, we observe that

u'. v.

v: v.

Then the induced maps

and G*A~A' induce maps of filtered, graded complexes:

F*(A(U.)) --> F'(G*A(U:)) -+ F*(A'(U'.)).

The homotopy h : U : x A [ I ] - + U . covering the trivial homotopy induces homotopies g'-lV(p Ix A[I] ~ g-iV(p). Therefore, the maps

A(g-~V(p), g-lV(p_t) ) -+ h*A((g' x I)-tV(p), (g 'x I ) - - l v (p_ t ) ) * t ~ ~ r - - 1 t--1 -+ p q A ((g • x)-~V(p), (g'• :2; A (g V(p), g V(p_x))

are homotopic. Finally,

restricts to give homotopies

U:xa[i] h, U.

V, x A [ I ] k �9 ) V ,

, -~1 , x , ~ [ I ] -,- g --(p) g-lV(~+l 1

Represent an element x of E~' q(g, A) by a cocycle in

Z~,'(g, A ) = Im(HP +q(g-XV(p+~), g-~V(p_~); A) -+ HP+~(g 'V(p), g-~V(p_l); A)).

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~o E R I C M. F R I E D L A N D E R

We must show that l~(x)--h*l(x ) lies in

B~'q(g ', A') = Ker(H~+q(g'-lV(p), , - av , . Hp+qt ,-l~z, g --(p_~), A') ~ ~g --(pl, g '-lVcp-2)); A')),

where h~ = hor : U: -+ U'. • A[1] -~ U. . This follows from the observation that the maps

s t - - I t A(g- a V(p + ~), g- ~V(p_~)) --> h A(g V(p) • A [I], g ' - xxr, --(p_2)A A [ I ] ) v . t ~ - -1 ~ A ~ t �9 t - - l ~ ) , ~ _ , - 1 V ~ -+ p q A (g V(n •215 : ~ ' " ~g --(p),g (p-2))

are homotopic.

The application of Lemma (3.2) which we envisage is the following: A is a locally constant, abelian sheaf on X and f : X-->Y is a map of locally noetherian schemes. We observe that A determines a local system on U . = r ~ ( U ) , U in Jx, whenever A is constant on each component of U0. Furthermore, if ~ ' ~ is a map in Jr, then the local system on U' determined by A is simply the pull-back of the local system on U determined by A. By an abuse of terminology, we shall denote by A each of these local systems determined by the sheaf A.

In the following proposition we verify that E p' q(g, A) for a Kan fibration g is the cohomological Serre spectral sequence for g and A. We first observe the following useful fact. Let g : U - + V . be a map of simplicial sets and let A be a local system on U. . For each simplex v of V , let

g - a ( v ) = V • and g - ~ ( b ) = U vXSk(p),

where Alp] ~ V . is given by v. Since

(g-~Vl,l, g-~Vip_t) ) -+ ~ (g-~(v), g-~(6))

is an excision map, we have

HH'(g-'(v), g-~(~) ; A) -~ H'(g-~V~.), g-~Vc._~; A)

where V + consists in the non-degenerate p-simplices of V.

Proposition (3.3). - - I f g : U. ~ V . is a ivan fibration and if A is a local system on U., then E~'q(g, A ) = ~ H q ( g - l ( v ) , A) and (El, dr) is the complex of normalized cochains on V

with coefficients in the local system v~Hq(g-l(v) , A). Hence E~'q(g, A ) = H ~ ( V , v ~ Hq(g-l(v), A)) => HP+q(U., A). Proof. - - Since g is a fibration, g contains a Kan fibre bundle h as a strong

deformation retract over each connected component of V. ([i6], ( i i . i 2 ) ) . Hence, g-l(v)-+A[p] is fibre homotopy equivalent to a product h- l ( v )=F• Therefore,

HP+q(g-X(v), g-~(6) ; A) -~ H '+" (F • sk(p)) ; A) U- Hq(F, A) Hq(F• A) ~- Hq(g-a(v), A).

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F I B R A T I O N S IN ETALE H O M O T O P Y T H E O R Y o !

Furthermore, g-l(div)~g-l(v) and g-l(sf)-+g-l(v) may be viewed as

~i oJ F x A [ p - - ~ ] --~ FXA[p] and F X A [ p + ~ ] ~ F x A [ p ] ;

thus, H'(g-l(v), A) --% H*(g-l(d~v), A) and H~ A) -% H'(g-~(sjv), A). We conclude that v~H~(g-l(v), A) is a covariant local system on V (cf. [7], App. II , (4.6) for more details).

As remarked in Construction (3. z), dl : E~'q(g, A) ~ E~+l'~(g, A) is the connecting homomorphism for

O --~ A(g- lVb+l) , g-lV(~)) -~ A(g-lV(~+l~, g-IV(,_l) ) -+ A(g-lVb), g-lV(~_l) ) --+ o.

To identify dx into some factor H~(g-~(v), A) of E~+X'~(g, A)---- H Hq(g-l(v), A), suffices to examine

o -+ A(g-l(v), g-l(~)) --~ A(g-~(v), g-l(iJ)) --~ A(g-X(~), g-l(U)) _+ o

it

where g-i(V) = U. vXA[p -t- x](p_x).

In this case, a cohomology class ~ in

HP +~(g-t(g), g-l(V) ; A) ---- ]-I H~'+'(g-l(d,v), g-t(d;v); A) O~__i.~ p + l

goes t o ~a=~. (--I)i0ig in HP+q+l(g-X(v), g-l(fi); A), where 0~ is defined by extending

a cocycle s in A~+q(g-l(d~v), g-a(d;v)) to some cochain Y in AP+q(g-l(v), g-l(d;v)), then restricting 0~ to a cocycle in AP+~+l(g-l(v),g-l(g)). Replacing g-l(v) by a strong deformation retract F • we readily check the commutativity of

HP+q(g-i(d~v), g-l(d;v); A) e , Hp+q+l(g_t(v) ' g_t(~); A)

H~(g-X(d~v), A) < Hq(g-l(v), A)

where the bottom row is induced by g-l(div) ~ g-l(v). We conclude that di : l - [ H q ( g - l ( v ' ) , A ) ~ H Hq(g-l(v),A) into some factor

v~+l Hq(g-l(v), A) is ~'(--1)r where the sum ~ ' is taken over these i, o < i < p - F t ,

i i - - - -

with d~v in V + . Hence {HH,(g-l(v), A), dl} is the complex of normalized cochains

on V with coefficients in v ~ Hq(g-t(v), A).

The map of spectral sequences EV'q(f;t, A) ~ Ev'~(f~t, A) will be our main tool in comparing the cohomology of ~(f/t) and I)(f~t).

297


Proposition (3.4). - - Let f : X-+Y be a (pointed) map of locally noetherian schemes and let A be a locally constant, abelian sheaf on X. For each ~ : U. -+V in Jt with A constant on Uo, Construction (3. i) provides a map of spectral sequences

E p, q(gr, A) -+ E p' q(g, A)

with common abutment, where gr : U ~ V is the fibre resolution of g : U - + V . Therefore, we obtain a map of spectral sequences with common abutment:

E~,q(f[t, a ) = l i m HP(V. v I > Hq(gr-l(v), A)) :~ HP+q(X, A) )

Ho Jf 1

F_~'q(f~t , A)= lim H'(H'+q(g-IVI,I, g-lV(,_x); A); dl) =~ H'+q(X, A) ~s

Proof. - - By Lemma (3.2), the commutative triangle

U . " - U r ' r

",,v./ induces the map EP'q(g ~, A) -~EP'q(g, A). Since homotopies in HoJt give homotopies of pairs of simplicial sets, the limit spectral sequences beginning with the E 2 term are well-defined by Lemma (3.2). By Proposition (3.3), E~'q(f[t, A) has its stated form. Finally, the abutment lim H*(U~, A)---lim H*(U., A) is isomorphic to H*(X, A)

) ) ,

Ho Jf HO Jf by Verd]er's theorem ([3], w V, App.).

Since any etale map of schemes U-~X with section s :X--~U splits as (U--s(X))HX-,--X, U. in A~ admits a unique splitting ([4], w 8). Hence, U--, is a disjoint union of copies of U + , the non-degenerate part of Uk, for o< k <n.

Lemma (3.5). - - Let Y be a locally noetherian scheme, let V be a simplicial scheme in A~ and let V.----~(V.). For each simplex v of V of dimension d, corresponding to a map v : A [ d ] ~ V , there exists a unique map g(v)~V, in A~ satisfying:

a) S(v)~V. is componentwise the identity; and b) ~ ( g ( v ) - + V ) = v : A [ d ] ~ V .

Furthermore, let f : X-+Y be a (pointed) map of locally noetherian schemes and let g : U . ~ V be an object of Jr" For each d-simplex v of V , define ~-a (v)=U xS(v) and ~-a(~)=U. xska_tg(v ). Then n(~,-t(v)) equals g-X(v) and v

V. rc(~-l(v), ~-l(~j)) -= (g-t(v), g-l(~)).

Therefore, for any locally constant, abelian sheaf A on X which is constant on Uo, H*(~-l(v), A) equals H*(g-l(v), A) and H*(~-l(v), ~-l(~j); A)=H*(ker(a( j - l (v)) -+ A(~-l(~)))) equals H,(g-t(v), g-t(g) ; A).

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FIBRATIONS IN ETALE HOMOTOPY THEORY 23

Proof. - - Properties a) and b) serve to define S ( v ) - + V . The equalities =g-t(v) and follow immediately from a)

and b), and directly imply the equalities

H*(g-t(v), A ) = H ' ( g - t ( v ) , A) and H ' (g - ' (v ) , g-~(~) ; A) = H'(g-~(v), g-t(~) ; A).

We proceed to analyze E~'e(f~t, A).

Lemma (3.6) . - - Let X be a scheme and let U.-+A[d] be aspecial map in A~ Let F be an abelian sheaf on X and let F-+I" be an injective resolution of sheaves. Then the bicomplex

ker{I '(U.) -+ I'(U._x_x sk(a))} ~[al

yields the spectral sequence

El' q= (ker(3~qF(U.) ~ o~fqV(U._x_X sk(d)))), :~ HP+q-a(X, F). a[a]

Proof. - - Given a scheme W etale over X, we denote by Z w the abelian sheaf on X given by Hom(Zw, F ) = F ( W ) , for any abelian sheaf F on X. We first verify that t-I.(ZK/Z0. • 7 ~ ) = ( Z x , d), the complex of sheaves which is Z x in dimension d and o

A[d]

elsewhere. I t suffices to prove that the canonical maps

z ./z0. • (Zx, a) AMI

induces isomorphisms H.(Z0./Z~. • ~-~)x-+ (Zx, d)x at every geometric point x of X. AM]

Since U. • x-~ Aid] and U._X_X sk(d) • x-+ sk(d) are special maps in A ~ (Sets), these are x a[aI x

fibrations with contractible fibre. Therefore, the maps

H.(Z~./Z~. • ~ ) , = H.(Z~. • • ,k(a)• -+ H.(A[d], sk(d)) = (Zx, d), ACd] X Aid] X

are isomorphisms.

We observe that ker(I*(U.)-+I*(U._x_x sk(d))) equals Hom(Zo. /Zo. • ~-~, I*). Therefore, a[a] ata--i

Hq(HP(ker(I*(U.) -+ I'(U.___x sk(d)))))

Aid] =I-Iq(Hom(Hp(Z0. ~_~_;kC~), I*))= iHq(X' F) if p = d i t o p , a!

so that the total cohomology of the bicomplex k e r ( I ' ( U ) - ~ l ' (Uo~Sk(d ) ) ) equals H ' - a (X , F). a~al

Since U _X_X sk(d) --* U. is componentwise the identity map, a[a]

Hq(ker(I*(U.) ~ I'(U.__x_x sk(d)))) = Hom(Z~./Z0. • ~-~, ~q(F) )

equals ker(affq(F)(U) --* ~q(F)(U._x_x sk(d))). atdl

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24 E R I C M. F R I E D L A N D E R

The proposition below relativizes Verdier's theorem (which can be obtained by setting d = o , V =A[o] , and f = i d :X-+X) .

Proposition (3.7)- - - Let f : X-+Y be a (pointed) map of locally noetherian schemes and let F be an abelian sheaf on X . Then for any V. in Jy, all d, q> o

lim IIHd+q(~-t(v), g-tQJ); F) -% HH~(vxX, F). ~o 9,~. v~ v~

Proof. ~ Let P be an abelian presheaf on X which commutes with direct sums: P ( U n W ) = P ( U ) • For any v in V +, ker(P(~-l(v)) -+ p(g--1Qj))) equals

ker(P(~-l(v) 2< A [d] • v) ~ P(~-l(v) _>< sk(d) • v)), since the components of ~-l(v) not in S(~) Y S(~) Y

~-l(~) are exactly thc componcnts of ~-l(v) • A[d] • not in ~-l(v)_• sk(d) • Let 8(v) Y S(v) Y

F-+I* be an injective resolution of abelian sheaves on X. Applying Lemma (3.6) with X replaced by v • and U replaced by ~-l(v) • A[d] • we conclude that the

Y g(v) Y

spectral sequence for the bicomplex ker(I'(~-~(v))-+ I*(g-~(~))) can be written

E~'q(~, F, v)-= (ker(o~f'qV(~-l(v)) -+ ~,~r ~ H~+q-a(vX X, F). Y

We define the map I IHa+q(g- l (v) ,g- l (~) ;F) -+~Hq(vxyX, F) to be the edge Vd Vd

homomorphism of the product spectral sequence l-[EP'q(g, F, v). Since homotopies v~

in Jf,~. between ~':~:~ induce homotopies between ~qF(~-i(v)) :~ JffqF(~'-l(v)) and ~fqF(.~-l(~)) ~ ~qF(~'- i (~)) , ~EP'q(~, F, v) is functorial on HoJf, v .. Moreover, if

Vd

g'-+~ is a map inJf,% and v~V +, then

K-'(v)_x ~[d] x v ~ g - ' ( v ) • a i d ] x v s(~) x sCv) Y

A [d] x v g

and g'-l(v)_Xsk(d)xv ~ g- l (v)xsk(d)xv S(~) Y / X

sk(d) • v Y

commute; hence, ~'-,,-~ induces an isomorphism on abutments for

HEp, ~(g, F, v) -+ IIE~, ~(g', F, v). v5 v5

It suffices to prove that lira ~E~,q(~, F, v )=o for q>o in order to complete the ~o Jf, v. vd

proof. Let x~, be an n-cochain in IIker(aC~qF(~-l(v)) -+ a~'qF(~-l(6))). For each ~o in v~

ker((o~qF(~-l(v)) -+ ~qF(~-l(~)))) .=o~qF(~-l(v).--~-l(~).) ,

let % :W~-->~-l(v),--~-l(~J), be some etale surjective map such that q~;(a,) -o in o~qF(W~). Since ~-l(v).--~-lQJ). embedsin U.=s(~) . such that g-l (v) . - -g- lQJ) , and

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FIBRATIONS IN F.TALE HOMOTOPY THEORY ~5

g - l ( v ' ) , - - j , - l ( 6 ' ) , are disjoint for v4~v' in V +, u% extends to an etale surjective map q~ : W = (u+ W,) ILl (U.--g-1 (V+),) _+ U., where g- t (V +). consists of the components of U,

mapping to some non-degenerate component of V + . Using Lemma (2.3), find a special map + : U~-+U such that U,~-+U. factors

through q~ :W-+U, and let ~' : U : - + V be the composition got~. By construction, +'(• = o in ~Ef'q(~ ', F, v)=~ker(~F(g'-l(V)vd ---> ~qF(g'- l(6)))"

As an immediate corollary of Proposition (3.7), we obtain the following comparison of simplicial and algebraic cohomology.

Corollary (3.8). - - Let f : X ~ Y be a (pointed) map of locally noetherian schemes and let A be a locally constant, abelian sheaf on X . For all V in J r , all d, q > o

lim Hd+q(g-tV(a), g-~V(a_l); A) -% I lHq(vxX, A). > Ho Sf, V. V5 V

Proof. - - By excision, Ha+q(g-XV(a), g-~V(a_~) ; A)=l-IHa+q(g-t(v), g-a(6); A). Apply Proposition (3.7). v?

Corollary (3.8) enables us to interpret E p' q(f~t, A) as a simplicially derived Leray spectral sequence, as explicated in Theorem (3.9) below. In conjunction with Pro- position (3.4), this enables us to readily compare the Leray spectral sequence with the Serre spectral sequence for f~.

Theorem (3.9). - - Let f : X-+Y be a (pointed) map of locally noetherian schemes and let A be a locally constant abelian sheaf on X . For the spectral sequence EP'q(fet, A) of Proposition (3.4), E~'q(f,t, A) equals HP(Y, Rqf.A). Therefore, EP, q(f~t , A) may be written

E$, q(f~t, A)= H'(Y, Rqf.A) => H'+q(X, A).

Proof. - - As given in Proposition (3.4),

E~' q(f~t, A) = li_>m H'(H p + q(g- 1V(p), g- 1V(p_t); A) ; d~). tto J f

Since J t ~ J r is fibrant with fibre Jr, re.,

lim H'(HP+e(g-IV(p), g-lVc~_l); A); dl) = lim lim H*(HP+q(g-IV(p), g-~V(~_l); A) ; dl).

As in Lemma (3.2), H'(g-lV(p), g-IV(~_l); A) determines a functor on HoJt, rr thus, E~' q(fet, A) equals lim~ H*( lim~ Hp+~(g-t Vt~), g- t V(p_ t),'A) ; dl)

HoJr Ho Jr, V. By Corollary (3.8) :

lim Ef'q(~, A) --% ~Hq(vxX, A). Ho~r,V" v~; Y

301


Moreover, d 1 equals lim dl(g), where Ho Jf, ~.

dx(g) : HHP+q(g-~(o'), g-a(6'); A) --% H HP+q+a(g-t(v), g- t (g ) ; A)

is the coboundary map for

o-+A(g-tV(p+~), g - tV(p)) -+A(g-tV(p+l), g - tV(p_ ~)) -+A(g- tV(p), g-~V(p_~) ~ o .

Let 3(g) be an " un-normalized " restriction of dl(g) :

~(~) : H HP+q(g-t(div), g-~(d~v); a) -+ HP+~+~(g-t(v), g-~(6); A). 0 ~ i ~ p + l

We view ~(g) as the coboundary in H ''~ cohomology for

o + i . ( g - , ( o ) , _+i.(g-t (v), g- t (~) ) ~ i . (~-~(6) , g-~(~)) -+o

where A-+I ~ is an injective resolution of abelian sheaves on X. As checked in Proposition (3.7), the associated E ~' q spectral sequences of these bicomplexes degenerate in the limit at lim E~ 'q. Therefore, lira 3(g) = 8 may be viewed as a map on abutments.

Ho Jf ,~,

Since the E q'v spectral sequences for I*(g-a(v), g-1(6)) and I'(~-x(6), ~-~(6)) degenerate at the E~' ~ level as checked in Lemma 3.6, we may view 8 as a map

II HP+q/I 'rZ H v+q O ~ i ~ p + l y y

We readily conclude that this map

II Hq(d,v~X, A) -+ Hq(v vx X , A) 0~<i~<p+l

is simply the alternating face map induced by the maps of schemes v--->div. For cr in I) d~a, where the ~ H q ( v ' ~ X , A ) , dlcr in Hq(v~X,A) is therefore the sum ~ ' ( - - ~ "

sum is taken over those i, o < i < p + I, such that div is in V + . We conclude that { lira HP+q(g-lV(p), g-IV(p_1); A); dr} is the normalized

H~ Jf, Vo

complex of the co-simplicial group: n ~ H q ( V . ~ X , A). By Verdier's theorem,

E~,q(fet, A) = lim HP(n~+Hq(V. yXX, A)) equals HP(Y, Rqf.A): the cohomology of Y HoJy

with coefficients in the sheaf associated to the presheaf S ~ H q ( S ~ X , A).

We remark that EP, q(f~t , A) can also be written

E2V'q(f0t, A ) = lim H ' (V. , v~R'f .A(@)=~H"+q(X, A), IIoJy

since Verdier's theorem asserts that

lira H P ( n ~ H ' ( V n ~ X , A) equals lim H ' (n~Rqf .A(V. ) ) . HoJy HoJy

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4. Applications t o a Proper, Smooth Map.

In this section, we apply Proposition (3.4) and Theorem (3.9) to the special case of a proper, smooth, pointed map of noethcrian schemes, f : X - - ~ Y . The Zeeman comparison theorem [i5] enables us to compare the cohomology of the geometric and homotopy theoretic fibres off0t. Under additional hypotheses on the fundamental groups nl((X~)ot) and nl(Y0t), we obtain the long exact sequence of homotopy pro- groups asserted in Corollary (4.8).

I f Y is a pointed, connected, noetherian scheme, the Grothendieck fundamental group n~(Y) of Y is the profinite completion of:~l(Y,t) ([4], (io. 7)). I f Y is in addition geometrically unibranched and if V is an etale hypercovering of Y with V,, noetherian for each n, then n,(V.) is finite for all n > I ([4], w Ix). For such Y, n,(Y0t) is therefore profinite; in particular, ~(Yet) equals ~I(Y).

Wc begin by observing (in Corollary (4.2)) that if f : X ~ Y is a pointed map of locally noetherian schemes with Y noetherian, we need only consider those ~ : U . ~ V . in Ho Jr with V, noetherian for each n > o. We shall frequently use this property of V to conclude that lim Hq(V., ~ ) = H~(V., lira A,).

Proposition (4.x). - - Let Y be a noetherian scheme and let V ' ~ V be a special map in

A~ with V , noetherian, n > o. Then there exists a map V'.' ~ V ' . in A~ such that

the composition V ' . ' ~ V ' ~ V . is special and V,]'-+V, is of finite type, n > o.

Proof. - - Let f : W-+Z be an etale, surjective map of schemes with Z of finite type over Y. Since f : W ~ Z is locally of finite type, there exists an affine open W w of W which is of finite type over some affine open Zw of Z for every point w of W. Since Z is noetherian, each Zw-+Z is of finite type, so that Ww-+Z is likewise of finite type. Since f : W-+Z is an open mapping, some finite union W' of W,/s satisfies the condition that f (W' ) = Z . We observe that W ' ~ W is etale and that the composition W ' - + W - ~ Z is etale, surjective, and of finite type.

To obtain V" . , we first apply the above observation to the map V0-+V 0 to obtain V0'. Having defined the k-th truncation, (V'.')Ik/-+(V'.) Ik), we obtain V~+ 1 by applying the above observation to

- - p - - _ _ _ _

Vk+l • _ (cosk~V'.')k + 1 ~ Vk+~ • (coskkV'.')k + 1, (c~ V;)k § 1 (e~ V.)k + 1

which is surjcctive since Vk+I-+V k el • (coskkV:)k+ I is surjcctive. By construc- (e~ V.)k. a

tion, (V~')Ik)-+(V'.) Ik) extends to (V")lk+a)~(V'.) Ck+~). By induction, we conclude that V~+ 1 Vk+ 1 • (cOskkV")k+l is finite type and surjective and V~,+l--~Vk+x is of

(c~ V.)k + t finite type.

Given a pointed map f : X---~Y of locally nocthcrian schemes with Y noetherian,

wc define various codirected and cofiltering categories associated to f. Let 'Jr bc the full

303


m m m

subcategory of Jr consisting of pairs U . ~ V . with V, noetherian for all n>o. Let ",It be the category whose objects are those of Jr and whose maps are simplicial squares. Let Ho'Jf and Ho"Jf denote the homotopy categories (with respect to pointed categorical homotopies of simplicial pairs) of 'Jr and "Jr respectively. Arguing as in Pro- positions (I. 2) and (1.4), we conclude that 'Jr and Jf are codirected and that Ho'Jf and Ho"Jt are cofiltering.

Corollary (4.2). - - Let f : X-+Y be a map of locally noetherian schemes with Y noetherian. Then Ho'Jt and HoJt are cofinal subcategories of Ho"Jt. Therefore, fet={g}noa; is canoni- caUy isomorphic to {g}ao, Jf in pro-3s

Proof. - - Proposition (4. I) implies that Ho'Jf~Ho"Jt is cofinal provided that HoJf~Ho"J f is cofinal. Given maps

u'. u.

v'. v .

in "Jr, we obtain as in Proposition (i-4) a map

v'.' v'.

v'.' v'.

in Jf such that the compositions are categorically homotopic. Therefore, HoJ f~Ho"J t is cofinal. Moreover, the cofmal maps Ho'J t~Ho"Jf and H o J f ~ H o " J t induce isomorphisms {g}rio,,af--~{g}Ho,af and {g}ao,,a/.-+{g}rloJf in pro-,Z/'o,p~i, ..

In the following proposition, we assume that 7~l(Ya)-~o. This simplifies the proof, while providing a good example of how Theorem (3.9) can be applied.

Proposition (4.3). - - Let f : X-+Y be a pointed map of locally noetherian schemes with Y noetherian. Assume that Y is connected and 7 h (Yet) = O. Let A be a locally constant abelian sheaf on X such that P,3f.A is constant on Y and (Rqf.A)y-~Hq(Xv, A) for all q>o , where X u is the geometric fibre o f f

Then (Xy)et-+[~(f~'t) induces isomorphisms for all q ~ o: ,~ , q A) H A).

Proof. ~ Since n l (V . )=o for all V. in Jy ([4], w IO), the local system v~Hq((g*)-l(v), A) may be identifed with its value, Hq([~(g'), A), on the distinguished vertex of V.. Since V. in 'Jr is of finite type in each dimension,

lim HP(V., Sq(~(g'), A)) = lim HP(V, nq(I~(g'), A)) > >

H o J f I t o ' a f

equals lim H~(V. Hq(i~(f:t), A)) = lim HP(V., HP(i~(f:t), A)).

> H o ' a f

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FIBRATIONS IN ETALE H O M O T O P Y T H E O R Y ~

Therefore, El"q(ff~t,A)~EV,~(f~t,A ) of Proposition (3-4) may be written

E,V,q(f[t , A) =HP(Yot, Hq(D(Zt), A)) :~ HP+q(X, A) l" F_~,q(f,t , A ) = H ' ( Y , t , (Rqf.A)v) =~ H'+ ' (X, A)

where

EC' (At, A)= lira a ) ) = lim (R'AA)y). Ho Jy Ho Jy

For each V. in J r , the composition

lim II Hq(I~(g',)A)~ lim H'((g')-lV(,), (g')-xVc,_l);A) no Jr, V. P Ho Jf, V.

-+ lim H q(g- ~V~), g- ~V(v_ 1); A) -~ lI H q(v ~ X, A) + YI Rqf. A (v)

is induced by the composition map of groups

~ q (Rqf. A)r lim Hq(I~(g'), A ) ~ lira Hq(I~(g), A ) ~ H (vo~X , A ) ~ H~ Jf, V. Ito Jf, V.

Therefore, the limit map �9 : EV'q(f[t, A)--*EP'q(f~t , A) is induced by the composition map of groups Hq(I)(f[t), A)~Hq(D(f~t), A)--*(Raf.A)v. Applying the Zeeman comparison theorem [I5] , we conclude that H~(D(f[t), A) ~ (R~f.A)v , all q>o. Applying Proposition (2.4), we conclude that Xot--*I)(f~t) induces isomorphisms H*(D(ff~t), A)~H'((Xu).t , A).

We recall that a map {X,} ~{Y~) in pro-J{" o is said to be a ~-isomorphism provided that the induced maps cosk~{X,}-+coskn{Yj} are isomorphisms for all n~o . A map

f : {X,}-+{Yj} in pro-~,T" 0 is a ~-isomorphism of connected pro-objects if and only if f . : 7rl({X,} )-~rq({Yj}) and for every rcx({Yj})-module M, H*({Yj}, M)--%H*({X,}, M) ([4], Theorem (4-3))-

The following lemma will enable us to relax the hypothesis on f : X ~ Y that rh (Y,t) = o.

Lemma ( 4 . 4 ) . - - Let f : X-+Y be a pointed, galois cover of connected, locally noetherian

schemes. For all n ~> 2,

f . : =,(Xot) ~=,(Y,t).

Proof. - - Let K denote the full subcategory of Ho Jy consisting of pointed, etale hypercoverings V. of Y such that Vo-+Y factors through f : X ~ Y . Given V in Ho Jy, V'.=cosk0X • V ~ V is special (where (cosk0X)k is the k-fold fibre product over Y

eoskeY

of X with itself). Hence, K-+HoJr is cofinal.

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30 E R I C M . F R I E D L A N D E R

Represent Yet as {~(V.)}K in pro-.,Y" o and let Zet={n(V. yXX)} K. Then

f, , :Xot~Yet in pro-~e" 0 factors as hetoget : Xet-+Zet~Yet in the obvious way. Since X/Y is galois, X • X ~ X is the trivial d-fold cover n X ~ X , where d equals

Y the degree off. For each V. in K and each k > o, there exists a commutative diagram with cartesian squares :

i

V , • ~ uX ~ X Y

V k ) X Y

We conclude that h~t : Z e t - - > Y e t , m q

Hence, ~(V. ~ X) ~ ( V . ) is a simplicial covering map.

given by {n(V. ~ X) ~ ( V . ) } K in pro-.:U0,~r, , induces isomorphisms h. : ~.(Ze, ) - ~ . ( Y a ) for n>2 .

For any abelian sheafF on X, get : Xet'-->Zet induces a map of spectral sequences

'E~,q-- -- ~lim HP(V.~x , ~q(F)) ~ HN(X, F) K

E~, '= lim HP(U., ~q(F)) ~ Hs(X, F) HoJ x

inducing an isomorphism on abutments. We observc that I imCP(V.• 9~~ ------> y K

equals li__~m 1-Ig~~ forall p, q>o , since (V.~X)p=HVp. Using Lcmma (2.3) , K

we conclude that 'E~ 'q=o=E~ ,q for all p > o , q:>o. Therefore, get induccs isomorphisms for any locally constant abelian sheaf F:

lim H*(V. ~X, F) = H*(Zet , F) --%H*(Xet , F) = lim H'(U., F). K HoJ X

Furthermore, {cosk0V.}itoJx-+{cosk0(V~X)}K is clearly an isomorphism; thus

~l(Xet)-~nl(Zet). Therefore, get:Xet--~Zet is a s-isomorphism. Wc conclude that the compositions ~.(Xa) "-~=n(Zet) --+~n(Yet) are isomorphisms, for n ~ 2.

In the following theorem, we eliminate the hypothesis that ~l(Yet) be o by examining the maps EP'q(f~'t' , A)~EP'q(f ' t , A) for each f ' : X'-+Y', the pull-back o f f by some pointed galois cover h' :Y '--+Y. Thus, we are led to consider direct limits of spectral sequences on the cofiltering category HoJ~" (Corollary (I.5)).

Theorem ( 4 . 5 ) . - Let f : X-+Y be a pointed map of noetherian schemes such that the geometric fibre X u is connected. Assume rq(Yet ) is pro-finite and Y is connected. Let A be a locally constant abelian sheaf on X whose fibre A x is finite with order relatively prime to the residue

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characteristics of Y. Assurae Rqf.A is locally constant on Y and

for all q >l o.

Then (Xv) ,t ~ ~ (fit) induces isomorphisms for all q ~ o:

(Rqf.A), ~ Hq(Xy, A),

Hq(I)(f;t), A)-~H~((Xu),t , A).

Proof. ~ Let h"-+h' be a map in ET /Y and consider the commutat ive diagram

t ! (Y4')~ > ( x T y , Y.,

t ( N l a > D(f,'() > (X;t)' > Y~t

r i t where f ' = f • X ' = X • and f " : X " ~ Y " . Since X v = X u =Xu is connec- Y

ted, X" and X' are connected; moreover =I(X")-+~I(Y") and =x(X ' ) -~ I (Y ' ) are surjective. Moreover, ~I (X")-+~I(X' ) and ~I(Y")-+~I(Y') are injective with cokernel Gal (X"/X ' ) = G a l ( Y " / Y ' ) . By Lemma (4.4) and the 5-1emma, [) ( f ; ( r) -+ [) ( f~'tr ) is a

H * r , l f ' ~-isomorphism. We conclude that (D(f~,), A) --% lim H (D(fl ,) , A), as well as n ' ( (xu) , t , A)-~ lira H'((X~).,, A). (ST/Y)

(ET/Y)

We consider the map ~ : l im (E p,q(g', A)--~E p' q(g, A)). Since lim = lim ]im lim ~oJ 7 ~oJ? ~T/YHoJy,~oJ/,,,~.

and since each V. in some HoJy , is of finite type in each dimension, ~ can be written

lim E~,q(g ", A) = lim lira HP(V., v -----& ~ > ), Hoa 7 E T / Y HoJy, [ I IoJ / , ,~7

l lim E~,q(g, A) ---- lira lim HP(V, v ~ Rqf'A(v))

]fitly ] IoJy,

lim Hq(g'-l(v), A)) =~ lim HP+q(X ', A) ) h~r/y

:~ lim HP+q(X ', A). ET/Y

For A satisfying the hypotheses of the theorem, Rqf 'A equals g"Rqf .A restricted to Y'; thus, Rqf 'A is locally constant and (t(qf.A)u-~ (Rqf'A)v ([3], XVI) .

Given any g in Ho J~, we can find g '-+g in Ho JT such that the induced systems v'w~Hq((g')-l(v'), A) and v '~Rq f .A (v ') are constant on V ' . Hence,

Hq(V., v ~ Hq((g') - ' (v), A)) He(V:, v' ~ Hq((g') - ' (v'), A))

4 Hr v ~ R~f.A(v)) Hq(V:, v' ~ R~f.A(v'))

3 0 7

3 2

factors through


H'(V: , Hq(I)(gr), A))

Hq(V:, (Rqf.A)v),

where the last pair is induced by the map of coefficients evaluated at the distinguished vertex v o of V.:

Uq(D(g'), A) -+Hq(D(g), A) -+ Hq(v 0 ~ X, A) -+ Rqf.A (v0).

Therefore, r may be written

l im E~,q(g ", A) = lim lim Hv(V. , lim Hq(~)(f~'(), A)) => lira HP+q(X ', A)

+ l im E~,q(g, A) = lim lim HP(V., (Rqf.A)y) ~ lira HV+q(X ', A) t~oa? ~ may, sT/~

with the map induced by

q , l im(Rrf~A) v (Rqf.A) v. Hq(l~(f[t) , A) = lim Hq(~(f, '(), A) ~ lim H ([~[fh), A) -+ q ' =

Applying the Zeeman comparison theorem [IS] and Proposition (2.4), we conclude that (Xy),t-+~)(ff, t) induces isomorphisms H*(i)(ff~t) , A) -%H*((Xu),t , A).

The " proper, smooth base change theorem " ([3], XVI , Corollary (2.2)) in conjunction with Theorem (4.5) immediately imply the following:

Theorem (4.6) . - - Let f : X - + Y be a proper, smooth, pointed map of connected noetherian schemes such that the geometric fibre X u is connected and rq (Yot) /s profinite. Let A be a locally constant, abelian sheaf on X whose fibre A S is finite with order relatively prime to all residue characteristics of Y .

Then (Xv)et-+I~(f;'t) induces isomorphisms H*(l~(f;t), A)-%H*((Xv)., , A).

Although the assertion of the following lemma is made for a pointed map f : X - + Y of schemes, the conclusion is a statement concerning only the associated pro-object, f~t, in pro-~0.p,lr .. Such pro-pairs arising from a pointed map of schemes are simpler to consider, since the pointed etale cover YH-+Y providcs a canonical representative of the covering space (Y,t)H of Y,t associated to a subgroup H of ~I(Y).

For any set L of primes, we denote by C L the class of finite groups G satisfying the condition that the prime factors of the order of G lie in L.

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F I B R A T I O N S IN E T A L E H O M O T O P Y T H E O R Y 33

Lemma (4-7). - - Let f : X-+Y be a pointed map of connected, locally noetherian schemes such that Y is noetherian and ~I(X) -+~I(Y) is surjective. Let L be a given set of primes such

that ~l(Yet) is in pro-C%. Let f~t~f~t be the canonical map of pro-3Uo.p~in , where (^) denotes completion with respect to Cr. ([4], w 5).

Then the induced map I)(ff~t)-+I)(a~ot ) in pro-J/o induces isomorphisms for all abelian A in CL, all q >_ o:

Hq(~(f/t), A) -%Hq(I~(f~'t) , A).

Proof. - - For each ~ of HoJ~, represent ~ in pro-.~0, pair, by {g' : U:-+V:}~t. I t t* ~ I t - - I r Then the maps {fit }~T/Y={g }HoJ 7 {{g },,}HoJ~--{fet }ET/Y induce

~): E~"q({f::}sr/y, A)~E"'({f~ ' tr }sT/Y, A).

As in the proof of Theorem (4.5)

E~' '({f~'(}ET/Y , A) = lim lim H'(V:, v ' ~ Hq((g")-* (v'), A)) tt0J] Ig

is isomorphic to

lim lim HP(V:, Hq(I)(g'r), A)) -% lim lim HP(V, Hq(I)(g'r), A)). m J/ Iz Hoa] I~

Since I)(g 'r) has L-primary finite homotopy groups, Hq(I)(g'r), A) is in C L. Therefore,

E~'e({~'tr}grm , A) --% lira lim HP(V., Hq(I)(g'r), A)) -% lim HP(V, Hq(I)(~r), A)) �9 ) - - - - - > �9

HoJ)7 Ig HoJ]

since V. is finite for n ~ o, if V. is in Ho'Jy.

We conclude that ~ : i~2, q({f~'(}ST~, A)---~I~P'q({f~'(}ET/y , A) may be written

p t lim H (Yet, Hq(~(d~',r), A)) ~ lim sP+q(x'~t, A)

p , lim H (Yot, He(i)(f~'(), A)) :~ lim H'+e(X ', A). ......._>

ET/Y

Since A is in CL, ~ is an isomorphism on abutments. As in the proof of Theorem (4.5), a map g"-+g' in ET/Y induces isomorphisms

H*(I)(f~'t'),A)-~H*(D(fg'(r),A). In order to conclude the lemma by applying the , ~ , r ,-~ * ~ ' , , r Zeeman comparison theorem, it suffices to verify that H (b(J~t), A ) ~ H (t?(f; t ), A)

whenever g"-+g" is a map in ET/Y. Consider the following commutative diagram in pro-oY'0:

J(fT; I - ') ) X " " > Y " et et

' ) �9 , , �9 y ' st

309


Since =I(Y' ) /=t (Y")==x(X') /=I(X") is in CL, ~ r ~ ; ~ and Y , , + - z , "~'et "~et are covering spaces / - . ~ , . /5~ ,~-; ([4], Theorem (4.x i)). Moreover, 7rl(Xet )firl(Xot ) --- 7rl(Yet ) firl(Yet ) =Trl(Y' )firl(Y'' ).

Using the 5-Lemma, we conclude that t)(f0t ')-+I)(f;t ' ) is a S-isomorphism; hence,

that H*(ik(f~'{), A) -+H (D(fet) , A).

Combining Theorem (4.6) and Lemma (4-7), we obtain the following homotopy sequence for a proper, smooth map.

Corollary (4.8). - - Let f : X-+Y be a proper, smooth, pointed map of connected, noetherian schemes. Let L be a set of primes not occurring as residue characteristics of Y and satisfying the condition that hi(Yet) is in pro-CL. Assume that the geometric fibre X~ o f f is connected and that

the L-completion ~I(X~) of ~l(Xv) is o. Then there exists a long exact sequence of pro-groups

. . .

where the maps ~,((Xv).t ) -+n,(X~t ) and ~.(Xot ) -+~,(Y,t) are induced by the L-completions of the etale homotopy type of the pointed maps i : Xu-+X and f : X--~Y.

Proof. - - By Proposition (2.2), i, :=,((Xv)et )-+n,(xot ) factors as

~. ((Xv) or) -+ ~. (I) ( f / t ) ) -+ n. (I) (f / t)) -+r:. (X,t).

Since Ex~~174 ") is a pro-fibre triple ([Io], Theorem (4.3)),

the triple D(dTo~t)-+XeZ-+Yo~--t induces a long exact sequence of pro-groups

To verify the corollary, it suffices to prove that the composition (Xu)et-~I)(f[t)-+I)(f~) is a S-isomorphism.

Recall that ~l(Xu) --~T:I(X ) -+7:1(Y ) -+e is exact ([8], w X). Since completion with

respect to C L is right exact, ~l(Xet) =~I (X) -+~I (Y ) ==l(Y~t) is an isomorphism.

Therefore, ~2(Y2t)--~,l(I)(/~t)) is surjective, implying that ~l(I)(/~t)) is pro-abelian.

Since ~l((X~)et) = o and ~:x(I)(f[t)) is pro-abelian, it suffices to check that the compo-

sition (Xv),t-+D(f[t~ -+I)(a~'t) induces isomorphisms H*(I~(f[t) , A)-%H*((X~)et , h) , any abelian group A in CT, ([4], Theorem (4.3)). This follows directly from Theorem (4.6) and Lemma (4.7).

5. A p p l i c a t i o n s to a V e c t o r B u n d l e m i n u s i t s o - sec t lon -

Given a locally free sheaf M on a scheme Y, the (algebraic) vector bundle V(M) associated to M is the spectrum of the symmetric Oy-algebra on M. In this section, we investigate the structure map f : V ( M ) - - o ( Y ) - + Y , where V(M)--o(Y) is the

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IqBRATIONS IN ETALE HOMOTOPY THEORY 35

vector bundle minus its o-section. We conclude that I)(ff0t) completed away from the residue characteristics of Y is a completed sphere.

A commutative triangle

Z ) V

of schemes is said to be a smooth S-pair of codimension r provided that i is a closed immersion, h and g are smooth, and for every geometric point s of S, Z s ~ V , is of codimension c. In particular, if M is a locally free sheaf of rank r on a scheme Y, then

0

y -',V(M)

is a smooth Y-pair of codimension r.

Lemma (5. x). - - Given a smooth S-pair

i

Z > V

\ s / of codimension c>o, with Z simply connected. Let L be any set of primes not occurring as residue

characteristics of S. Denote V - - Z by X. Then for any abelian group A in Cr., there exists the foUowing " Gysin " long exact sequence

in cohomology:

. . . ~H"-2*(Z, A) -+H"(V, A) ~ H " ( X , A) -+H"-2~+t(Z, A) ~ . . . .

Proof. - - Consider the Leray spectral sequence for the map j : X ~ V and the

constant sheaf A on X:

E~,q ---- HP(V, Rqj.A) =~ H p + q(X, A).

Recall that for j : X - + V associated to a smooth S-pair as in the present situation, j .A equals the constant s h e a f A on V; R q j . A = o for q4:o, 2c--I ; and i*(R2~-lj.A) is locally isomorphic to A on Z ([3], XVI, Theorem (3.7)). Since Z is simply connected, i'(R3~-lj.A) equals the constant sheafA. Hence R2~ A----i,A ([2] ,The-

orem (2.2)). Thus E~' q reduces to two non-vanishing rows: H'(V, R~ = H*(V, A) and H*(V, R2"-lj.A) --% H*-~r I.A) --% H*-2~+x(Z, A). This spectral sequence

therefore reduces to the asserted long exact sequence.

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3 6 E R I C M. F R I E D L A N D E R

The following propositi on verifies the cohomologi cal condition s on V (M) -- o (Y) ~ Y required in order to employ the simplicial Leray spectral sequence.

Proposition (5.2). - - Let Y be a connected, noetherian scheme; let M be a locally free, rank r sheaf on Y; and let f : X = V ( M ) - - o ( Y ) ~ Y be the restriction of the structure map g : V(M) -+Y. Let L be the set of primes not occurring as residue characteristics of Y. Then for any abelian group A in CL, Rqf.A is locally constant on Y; and (R*f,A)u~Hq(Xu, A) for all q ~ o , all geometric points y of Y .

Furthermore,

IA / f q = o , 2r--I Hq(Xu, A) = ~ o otherwise

for any abelian A in CL, any geometric point y of Y. Proof. - - We view f : X - + Y as the composition ofg with j : X ~ V ( M ) . Consider

the spectral sequence for the composite left exact functor fo and the constant sheaf A on the etale site of X:

F_~.' ---- R'g.(Rtj .A) ~ RSf, A.

As in the proof of Lemma (5. I), R~ = A on V(M) ; Rtj, A = o for t * o , 2r--I ; and o*(R2r-lj, A) is locally isomorphic to the constant sheafA onY. Since g : V(M)-+Y is acyclic for L (g is smooth with L-acyclic fibres), we conclude that R ' g , R ~ for s + o. Furthermore, for s 4: o,

R 'g , (R 2r- lj, A) = R'g,(o,o*R 2r- lj.A) = Rfid . (o 'R2r- l j , A) ---- o.

Hence, R2"-lf, A=g,(R2"-Xj, A) = (goo),o*(R2r-lj , A) is locally isomorphic to o * A = A on Y, whereas R~ =g, ( j ,A ) = A. Thus, Rqf, A is locally constant on Y for all q_~ o.

Since (PJf.A)u-%Hq(X,, A), where 0 denotes the strict local ring at a geometric point y, it suffices to prove that the natural homomorphisms H~(X,, A)-+Hq(Xu, A) are isomorphisms to complete the proof. The residue map Y-+0 induces a map of Gysin sequences for the smooth pairs of codimension r given in Lemma (5. i) :

2 ) /A~ : ( V ( M ) ) v 1) > /A~ = ( V ( M ) ) o

\ / \ / y 1)

Applying the 5-Lemma, we conclude that y---~t) induces isomorphisms

q Hq(X,, A)--->H (Xy, A)

for all q ~ o, all geometric points y.

Proposition (5.2) provides the hypotheses needed to prove the analogue of Theorem (4.6) and Corollary (4.8) for f : X = V ( M ) - - o ( Y ) - + Y .

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Theorem (5-3). - - Let Y be a pointed, connected, noetherian scheme such that %(Yot) is pro-finite. Let M be a locally free, rank r>__ 2 sheaf on Y. Let f : X = V ( M ) - - o ( Y ) ~ Y be the restriction of the structure map g : V(M) -+Y, pointed above the pointing of Y. Let L be some set of primes excluding all residue characteristics of Y and let (^) denote completion with respect to CL.

Then there exist g-isomorphisms in pro-~o:

s 2,-1_+ (x,).t a,d

Furthermore, i f =x(Y) is in pro-CL, there then exists a g-isomorphism consequently, there exists a long exact sequence of pro-groups:

�9 -. -+ r~.((Xu),t)-+=.(Xot) '+~.(Yet)-+=.-l((Xt,)ot)~-. .

A A

( x , , ) �9 e

A m A A

where the maps %((Xv),t ) -+ 7:,(Xet ) and rr,(X,t ) ~ 7~,(Yet ) are induced by the L-completions of the etale homotopy types of the pointed maps i : X u ---> X and f : X ~ Y .

Proof. - - To verify the existence of a S-isomorphism $2'-1 ~ (Xv)ot , it suffices to

verify that 7h((Xy), 0 = o and that

Hq ((Xv~----) et, Z) = 17. ~ q = 2 r - - I q 4= o, 2r-- I ([4], Corollary (4- 15)).

Yet 'Tq(Xv) ==I(V(M)u ) =Tq(/k~) by t h e " Purity T h e o r e m " ([3], XVI, Theorem (3.3)),

so that 7q((Xv)ot ) = o . Serre class theory implies that H~((Xv)ot, Z) is in pro-C L for

all q>o. Since H*(Xv, A)=H*((Xu)ot, A ) for abelian groups A in CL, Proposi- tion (5.2) and the universal coefficient theorem imply that

Hq((Xu)et ,z) = q 2r--I q4=o, 2r--I

as required.

To verify that (Xu)et-+ D(ffot) induces a g-isomorphism (Xu)~t-+ [~(ff~t), it suffices

to prove that =x((Xu), t)=o==x(i)(ff , t)), and that for any abelian group A in CL, H*(I~(f/t) , A) --% H'((Xw),t , A).

Observe that if Y" -+ Y is a pointed, galois cover in ET/Y, then

X' -=-V(M ~ d~w)-- o(Y' )

f l

y ,

> V(M)- - o(Y)

f

~. Y

313


is cartesian. V(M) --,- Y Applying the 5-Lemma plus Lemma (4-4), we conclude that the square

Furthermore, since ~l(X) --% ~I(V(M)) by the Purity Theorem and since has a section, ~I(X) ~ =x(Y) is surjective; therefore, X ' is connected.

( X ; ) 0 t ~ ,

1 1 ,

a #-isomorphism for any map commutes in pro-~U0, with I)(f/t ~) -+ I)(ff~t) in ET/Y.

y t --?~ y

We consider the long exact sequence of pro-groups

. . . . .

In particular, {n~.(Y;t)}~,T~ nl(I~(f~))~{nl(X;t)}ET/y'-" e is exact, since

{~l(Y;t) }V,T/Y = {z~I(Y')}ET/Y = e.

For any Y ' ~ Y in ET/Y, f ' : X ' -+Y' is i-aspherical for L; therefore, wl(X') =wx(Y').

Since (^) is right exact, we conclude that {n2(Y,t)}ST/y~ ~l([)(f[t)) is surjective; thus

wl([)(ff, t)) is pro-abelian. We conclude that it suffices to prove that (Xu)0t~ ~(ff, t) induces isomorphisms for any abelian A in C L : H*(~(f/t), A) --% H'((Xy),t, A). This is immediate by Theorem (4.5) and Proposition (5.2).

Finally, to conclude the existence of the asserted long exact sequence provided ~I(Y) is in pro-CL, we apply Lemma (4.7) as in the proof of Corollary (4.8) to prove that

the composition (X~),t-+ t)(f[t ) -+ I ) ( f~) is a $-isomorphism.

6. Adams' Conjecture.

In this section, we provide a completion of D. Quillen's sketch of Adams' conjecture for complex vector bundles [I3]. The arguments and results of Q.uillen's discussion are freely employed, together with Theorem (5.3) of the previous section. An additional ingredient is D. Sullivan's observation that pro-finite pro-objects of the homotopy category admit inverse limits.

Let ~ denote the homotopy category of topological spaces having the homotopy type of C-W complexes and let [ . , . ] denote maps in .~ . We recall that any pair in . ~ is represented by a Hurewicz fibration (namely, the mapping path fibration of any representative), unique up to fibre homotopy equivalence ([6], Theorem (6. x)). Thus, the fibrewise join E* E' -+ Y of pairs E ~ Y and E' ~ Y in 3~ is well-defined up to

Y fibre homotopy equivalence [9]. Moreover, the fibre ~(E -+ Y) of a pair E -+Y in o'V

314

FIBRATIONS IN E T A L E H O M O T O P Y T H E O R Y 39

is thereby defined up to homotopy equivalence; furthermore, t ) ( E ~ Y) is an object o f ~ [II] . We use [[.[[ to denote the geometric realization functor from (pointed) simplicial sets to ~ . I f S. -~ T. is a Kan fibration ofsimplicial sets, then [I I)(S.-,'-T.) II is isomorphic to D([IS [ l ~ lIT. I[) in Jt ~ [12].

We begin by recalling the representability of pro-finite inverse limits of connected objects of 3(f ([i4] , Propositions (3. i) and (3.3)). For the sake of completeness, we include Sullivan's proof.

Proposition (6. x) (Sullivan). - - Let {X,} in pro-~ satisfy the conditions that X, is (arcwise) connected and ~r,(X~.) is finite for each n> I, i in I. Then the functor

l im[ X~.] : ~ o _+ (Sets)

is representable in . ~ by a C-W complex, denoted lim X~. Proof. - - A functor F : ~ - + (Compact Hausdorff spaces) is a " compact repre-

sentable functor " provided that the composition with the stripping functor

(Compact Hausdorff spaces) ~ (Sets)

is representable in ~ by a connected C-W complex. Using Brown's representability axioms [5], we verify the following: if {F~}~ is a system of compact representable functors indexed by a cofiltering indexing category I, then lim Fi is a compact repre-

*(

sentable functor. Since lim< commutes with inverse limits, tim F~(VYj) equals ~. lim+__ Fi(Y~), where VY~

is an arbitrary wedge product of connected objects o f ~ (each pointed by a non-degenerate base point) ; furthermore, by the compactness of F~, hm F~(Y) maps surjectively onto lim F~(Y,), where Y , -+ Y satisfies the condition that sk,(Y,) -~ sts(Y ) in ~ for any

non-negative integer n. Let Y-- - -SuT and let Z = S n T be given in ~ , with Z a subcomplex of both S and T. Let lim s, x l i m t~ be an element of lim F~(S) xli+_m_m F,(T)

K------ <

such that the image of lim s~ equals the image of lim t i in lim F~(Z). Let Ci denote K-- - - - ~ ,(

the closed subset of Fi(Y) given as the inverse image of s~xt~ in F~(S)xF~(T). Since the inverse limit of non-empty compact sets is non-empty, there exists an element h m y l in hm C,r F,(Y) which maps to lim s~xli._mm t~.

Thus, it suffices to prove that { [ , X~.]}~ is a system of compact representable functors. Observe that if Yc is a finite subcomplex of Y, then [Y~, X~] is a finite set. For any two complexes X and Y, the natural map [Y, X] ~ lira [Yc, X] is surjective,

c

where Y~ runs through the finite subcomplexes of Y. We use the finiteness of homotopy classes of homotopies Y, X I ~ X~ restricting to given maps on Yc x o and Yc x I in order to verify that [Y, Xi. ] ~ hm [Y~, Xi. ] is injective. I f f , g : Y z~ X i map to the

c

same element of lim [Y~, X~.], we obtain an inverse system of non-empty finite sets of ,<......_

c

homotopy classes of homotopies between the restrictions o f f and g. An element of the

315

4 ~ E R I C M . F R I E D L A N D E R

inverse limit yields compatible homotopies which patch together to yield a homotopy between f and g.

We conclude that [Y, X~.]-----fim[Y~, X~.] admits the structure of a (totally c

disconnected) compact Hausdorff space. We readily check that any Y' -+ Y induces a continuous map [Y, X~.] ~ [Y', X/.]. Moreover, whenever X~.-+ X~ in {X~}i, then [Y, X~.] ~ [Y, X~] is continuous for any Y. Hence, { [ , Xi.]} ~ is a system of compact representable functors.

This inverse limit of pro-finite pro-objects preserves fibrations in the following sense.

Lemraa (6 .2) . - Let {E,---~Y~} I be an object of pro-.Yt~ such that E,, Y~, and

I?(E~-,'-Yi) are connected and %(Ei) and %(Y~) are finite for every n 2> I, i in I. Then

I)(li~__m E,--~ tim Y~) --~ l~m I)(E, ~ Y~) in off.

Proof. - - By autoduali ty of finite abelian groups, lim is exact on functors from I to (Finite abelian groups). Hence the long exact sequences associated to the triples

I)(li~m E, �9 lim Y~) ~ lim E, , l im Y~

I)(E, �9 Y~) , E, �9 Y,

imply isomorphisms %(I)(limm E,-+ tim Y~)) --% %(tim ~(E,-+ Y~)) for n 2 2. For n = I, we let E~ -+ Y~ denote the pull-back of Ei --~ Y~ to the universal covering

space Y~ of Y~. Exactness of lira on finite abelian groups and left exactness on arbitrary (

groups imply the exactness of lim ~(E~) -+ lim n2(Y~) --~ lim ~I(I?(E~ -+ Y~)) -+ lim nl(E~). Upon applying the 5-Lemma to the commutat ive diagram

~2(IimE~) , n2(lit.~nY~) , rh(D(li~mE~ �9 li+~mY~)) �9 ~l(l~mE~) �9 o

lim ~2(E~) �9 lim 7r2(Y~) , m rrt(I)(E ~ �9 Y~)) , lim r:x(E,)

we conclude that ~l(D(li~_m E" -+ lim Y~)) --% lim 7h(I)(E" ~ Y~)). Since

t)(limE~ , l imY' ) , lim I)(E~ , Y~)

commutes in off,

316

I)(l~m E~ , lim Y~) > lim I)(E, , Y~)

I)(lim Ei-+ lim Y~) -+ lim I)(E,-+ Y~) is an isomorphism in ,,~f.

F I B R A T I O N S I N ETALE H O M O T O P Y T I I E O R Y 41

The following lemma will enable us to define the fibrewise join of " L-completed

sphere fibrations. "

Lemma (6.3). - - Let (^) denote completion with respect to the class Cr~ of finite groups with prime factors in a given set L of primes. Let S and S' be m and n spheres, with m, n > I, respectively. Then the natural map S , S' ~ lim S �9 lim S' induces a S-isomorphism <------ <

(S �9 S')^ -+ (tim g �9 lim S')^.

Proof. - - Since the join of pathwise connected spaces is simply connected,

it suffices to prove that for all qeL, S �9 S' ~ lim g �9 lira S' induces isomorphisms < - - - -

S , Z/q) -% H*(S * S', Z/q). We recall that for any pair of connected spaces H'(l(im S . tim ^'

Z and Z' (cz , z) x (cz ' , z ') = ( c z x cz ' , z , z ')

where CZ denotes the cone on Z. Furthermore, we recall that H'(l im S, Z/q) =H*(S, Z/q)

([I4], " complement" Theorem (3.9) for m > I ; for m = I, use the fact that Z and lim 7. <

have same Z/q cohomology), thus finite. Applying the Ktinneth theorem, we

S ,Z/q) is isomorphic to conclude that H*(Clim S, lim S;Z/q) | S',li_m ^'"

H'(C l(im g • Cli+_m_m S', lim g �9 lim S'; Z/q). Therefore,

s , Z/q) H'(l im S, Z/q)@ H'(l(im ^'

s , z/q) U'+l(li+_ m g , lira ^'

~ , H*(S, Z/q) | Z/q)

H ' + I ( S , S', Z/q).

I f {Xi}-+ {Yj} is a map in pro-o-gf with {Yj} 1-connected, then

is a S-isomorphism in p r o - ~ ([4], Theorem (5.9)). Let Y in o~ be i-connected. I f E ~ Y and E' -+ Y are pairs in Jt ~ having fibres

lim S ~'-t and lim S "-a respectively, then ( (

Ig(E ~, E' --~ Y)^ = (lim S m~-I �9 lim~ S%~--t) ̂ ~ I?(E* E' -+ ~r)

is a S-isomorphism. By Lemma (6.3), (Sin+"-1) ̂ -+I ) (E . E' ~ Y ) ^ is a S-isomorphism. Y

Applying Lemma (6.2), we conclude that lim E �9 E' -+ lim Y has fb re lim(S'~+"-l) ̂ . < Y < <

Definition (6.4). - - Let L be a set of primes, (^) denote completion with respect to C L. A I-connected C-W-complex Y is L-good provided that the canonical map

Y -+ lim Y in o~ is an isomorphism. ,(

317


I f Y is L-good and m > i, SF~,(Y) is defined to be the set of fibre homotopy equivalence classes of Hurewicz fibrations with fibre lim S "-1. The symmetric operation

<

+ : SF~(Y) x SF.L(Y) -+ SF~+.(Y)

is defined by sending p : E - + Y , p' : E ' - + Y to the fibration p + p ' : f i m E , E ' ~ Y

induced by E �9 E' --+ Y. Y

If Y' is a I-connected, finite type C-W complex (i.e., H,(Y') is finitely generated

for each n), then for any L, lim Y ' = Y is L-good, where (^) is the completion with K----

r e s p e c t to this L ([x4], " complement " Theorem (3.9)). The following proposition introduces the monoid of stable, L-completed sphere

fibrations over an L-good complex.

Proposition (6.5) . - - Let Y be an L-good complex, let e, : lim S" - 1 • y ~ y denote the trivial fibration of SF~(Y), and let ~, denote

+ e . : sv (Y) - ,

Then {SF~(Y), r is filtering, with set-theoretic direct limit denoted SFL(Y). Moreover,

+ : SF~(Y) • SF~(Y) -+ SF~+.(Y)

provides the set SFL(Y) with the structure of a commutative monoid, natural with respect to homotopy classes of maps X - + Y of L-good C-W complexes.

Proof. - - The projection maps for ((tim S " - l * f i m S "-1) • induce a fibre

homotopy equivalence Hm((fim S"-/-'~. lira S "- t ) • -+ lim S~s • lira ~r over lira ~r < < < <

by Lemma (6.2) and the fact that I?(E --> Y)^ -+ I?(E -+ ~r) is a IMsomorphism. Hence e,(em) equals era+ . . To prove that {SF~(Y), e,} is filtering and that {SFL(Y), + } is a well- defined commutat ive monoid, it suffices to check the associativity of + . There are

limC i,n z , ^ and E , V : , im(Z. V : , natural maps Ey*E ' . < .< Y Y . Y

covering Y. By examining the induced maps of fibres, we conclude that (p + p ' ) + p " and p + (p '+p" ) are fibre homotopy equivalent to l im( /~*/ /* / / ' ] ^. < - - - y ~ y - - -

Let f : X - + Y represent a homotopy class of maps of L-good complexes, let p : E -+Y be in SF~(Y), and let p ' : E'-->Y be in SF~,(Y). Using the natural maps

( E ~ X ) , (E'yxX)-~-lim((E• Y * (E' yxX)) ^ and ( E . E ' ) • + l i m E . E ' • < Y y we

readily conclude that ( f f p ) + (f*p') is fibre homotopy equivalent to f * ( p + p ' ) . Hence, f induces a homomorphism SFL(Y) -+ SFL(X).

We recall the C-W complex Ba, the classifying space for sphere fibrations. For a finite dimensional C-W complex Z, the group [Z, Bo] is the group of stable fibre homotopy equivalence classes of sphere fibrations over Z. The following proposition

relates [Z, Bo] to SFL(lim Z).

318

FIBRATIONS IN ETALE HOMOTOPY THEORY 43

Proposition (6 .6) . - - Let Z be a I-connected, finite C-W complex, let L be a set of primes,

let (^) denote completion with respect to L, and let Y = l i m Z. The association 0flim/~ : tim E ~ Y to a sphere fibration p : E-+ Z defines a homomorphism of monoids

0r, : [Z, Bo] ~ SFT'(Y), any set L of primes.

Furthermore, i f p : E--~ Z is a sphere fibration in the kernel of OL, then the order of p in [Z, Bo] has no prime factor in L.

Proof. - - Let p : E ~ Z , p' : E'---~Z be sphere fibrations over Z, inducing a (not

necessarily unique) map p . p ' ~ f im/~ , tm/~ ' . Applying the Ktinneth formula to the

map on the cohomology of the fibres, we conclude that the induced map

fimpz*p' ~ fim/~ + t m / ~ '

is a fibre homotopy equivalence by Lemma (6.3). Moreover, if p : S ~ - I • is the trivial S "-1 fibration, then the natural map tim ~-+e m is a fibre homotopy equivalence of fibrations over Y. Hence, 0 L is defined on stable classes and is a homomorphism of monoids.

Let p : E - + Z represent a class in the kernel of 0 L. We may assume that there

exists a map lim E--~ lira S ~" whose composition with the inclusion of the fibre < <

lira S % --~ lim E is an isomorphism in ~,~. < K - - - -

O b s e r v e that [E, S %] equals [E, S %] equals [tim E, S %] since E is L-good, which

by definition equals [l<im E, lira fire]. Similarly, [~m, g,,] equals [tim fire, Hm ~,n]. A ^ ~ A

8 m We conclude that [E, S m] [ , S m] is surjective, where this map is given by composi-

t ionwi th the fibre inclusion S"-+F, i n p r o - ~ . Therefore, (E, S %) -+ (g~, 8") is likewise surjective, where ( . , . ) denotes the abelian pro-group of stable maps in p r o - ~ ([4], w 7).

Furthermore, (E, S%)=(E,S~) ^ and (S %, S%)=(Sm, S " ) ^ = Z ([4], Corollary (7-5))- In other words, the inclusion of the fibre S'~-+E induces a homomorphism (E, S =) -+ (S ~*, S") whose cokernel has L-completion o.

For some n~>o, there thus exists a map S"E-+S "+~ whose composition with S"+m-~S"E is a map S"+"-+S "+" of degree k, with k having no factor in L. By Adam's " mod .k Dold Theorem " ([I], Theorem (I. I)), applied to ( S " - I • �9 E -+ S " + " • we conclude that p : E - + Z represents a stable class in [Z, Bo] whose order has no factor in L.

The following proposition, proposed by Quillen ([I3] , w io), will suffice to prove Adams' conjecture. We recall that an algebraic variety V over the complex numbers can be triangulated; thus if V is projective, it is a finite G-W complex.

Proposition (6.7) . - - Let p be a prime number, let L be the set of all primes except p, and let (") denote completion with respect to C L. Let R denote the local ring given as the strict loca-

319


lization of Z at p, and let k denote the residue field of R. Let Ya be a scheme projective and smooth over Spec P,., with generic geometric fibre Yc over Spec C (C =complex numbers) and special geometric fibre Yk over Speck. Let Y,~ denote the " classical " topological space of complex points of Yc. Let Kt~ ) denote the Grothendieck group of complex vector bundles over Ya and let K=Ig(Y.) denote the Grothendieck group of locally free, finite rank sheaves over Yo.

I f Y~l is i-connected, then there exists a commutative diagram of monoids

el j* i* K~v(Yo,) < K'~(Yc) < K'~(YR) > K'~.(Yk)

SFL(Iim#Z,~) ~-- SFL( imll?c, t[]) SF '(limlleZR,.[]) SFL0i _ml[ rk, 0tII)

where J : Kt~ ) ~ [Y01, Bo] is the usual J homomorphism, where

J . (E.) = lim [l (V(E.)--o(g.))~t 11 li. _m II ?..,t II for E. locally free, of rank >--2 over Y. (see Theorem (5.3)), and where the lower horizontal arrows are induced by the isomorphisms in pro-~,

Yoi 5 11Yc, ot]l L [I ~R, otl[ L 11 ~k,,tI[. Proof. - - To prove that J . extends to a homomorphism on Kalg(Y.), it suffices to

prove that J . is additive with invertible image. For J c , these properties and the commutativity of the left-most square follows from the observation that ~*Jc(Ec) -= 0LoJo cl(Ec) in SFL(li+_m_m ~r ) , by the "generalized Riemann existence theorem" ( [4], Theorem (I 2.9)). For JR, we observe that f fJR(ER)=Ja( j*ER) in sFL(~m[l~c,.ll) by employing the proper, smooth base change theorem and lemma (5-I). Similarly, the proper base change theorem implies that the right-hand square commutes, provided ~ extends to a homomorphism on K~lg(Yk). There remains to prove the following:

i) for any locally free, rank > 2 sheaf E k over Yk, J(Ek) is invertible, and 2) for any short exact sequence

__ "r'~ t . _ . _ "t'~ lr'3 t l ( , ) o

of locally free, rank -->2 sheaves over Yk, J(E~) equals Jk(E~)+~(E~ ' ) . Let ( .) be a short exact sequence as above such that E k is generated by its global

sections. Let n be the rank of Ek, m be the rank of E~', and let N be an integer such that E is generated by N global sections. Let D~. denote the 2-stage flag manifold of type N, n, m. Then D R is projective and smooth over Spec R and D01 is x-connected.

Hence, JD : Kalg(DR) -+ SFI'( lim l] Da..t II) is well-defined. Moreover, ( ,) arises by pull- back from the universal short exact sequence over D R. Hence, Jk(E,)=,#r162 ')

320

F I B R A T I O N S IN ETALE H O M O T O P Y T H E O R Y 45

A and each of ~ ( E , ) , Jk(E~), and Jk(E~') are invertible, since Jk o i *= i 'o JR on actual bundles.

More generally, any locally free, finite rank sheaf E k over Yk is embedded (locally as a direct summand) in bL | where L is a very ample bundle on Yk and a, b are positive integers. By the above argument, ~ ( E , ) is invertible.

Finally, given any short exact sequence ( .) of locally free, rank > 2 sheaves over Yk, there exists a commutat ive diagram with exact rows and columns:

o > E~

o ~ E~

o o

T T F 3 = F 8

T T > F 1 > F 2 ) O

T T tt ' E k ,E~ , o

o o

with F x and F 2 generated by global sections. Hence

~(Ek) = A(Fx)-- ~(F3) = (Jk(E~) q- ~(F2) )- (Jk(r2) --J(E~')) t t! = ~ ( E k ) + ~ ( E k ).

In conclusion, we complete Quillen's sketch of Adams' conjecture.

Theorem (6 .8) (Adams' Conjecture). - - Let E be a complex vector bundle over a finite complex X and let k be a positive integer. For some positive integer n, the stable sphere fibration

associated to k"(+kE-- E) in Kt~ isfibre homotopically trivial. Proof. - - It suffices to prove the theorem for the canonical m-dimensional quotient

bundle O over the Grassmannian G of complex N planes in m + N space (for arbitrary m , N > I ) and for k = p , a prime. Let L be the set of all primes excep tp and let R denote the strict localization of Z at p.

Let G z denote the Grassmannian scheme over Spec Z representing the functor " isomorphism classes of m-dimensional vector bundles with m + N generating sections. " The pull-back of G z to Spec R, denoted GR, satisfies the hypotheses of Proposition (6.7)- Let Q.R denote the canonical locally free, rank m sheaf over GR; Q is the topological vector bundle over G---G~I associated to Q.R | O~ c over G c.

OGR

To verify the theorem, it suffices by Proposition (6.6) to prove that

O r o j ( ~ p ( Q ) - Q ) = o in SFL(li~mG). By Proposition (6.7) it suffices to prove that

Jk(d/ ' (Qk)--Qk) -= o, or equivalently that ~( t~ ' (Qk) ) = A ( Q ~ ) , in SFL(lim__m 111~k, et 11).

321

46 E R I G M. F R I E D L A N D E R

We recall that hb~(Q.k)=0~) in K~'gtc'-',~v, where O~P)is the pull-back of Qk by the Frobenius map q~ : G k ~ G k. Furthermore, there exists a purely inseparable map

Spec(Symo~k(Q.k) ) -=V(Q~) -+ v(o~)) = Spec(Sym~o,(O~)))

of schemes over Gk, given in local coordinates by (Z~) goes into (Z~). Therefore, the induced map in pro-OF 0 (V(Q,)--o(G,))o t-+ (V(O~))--o(Gk))o t is an isomorphism of pro-objects over (G,)~t ([8], Theorem (4. xo)). By definition of ~ , we conclude that

REFERENCES

[I] J. F. ADAMS, On the groups J (X) , I, Topology, 2 (I963) , I81-I95. [2] M. ARTIN, Grothendieek Topologies, Harvard Seminar Notes, i962. [3] M. ARTXN, A. GROT~Z~DmCK and J.-L. VF.RDmR, SOninaire de g~omgtrie algtbrique ; Cohomologie ~tale des scMmas,

x963-x964, notes mim~ographi~es, I.H.E.S. [4] M. ARTIN and B. M~zuR, Etale Homotopy, Lecture notes in mathematics, n ~ Ioo, Springer, x969. [5] E. BROWN, Cohomology theories, Ann. of Math., 75 (x962), 467-484 . [6] A. DOLD, Partitions of unity in the theory of fibrations, Ann. of Math., 78 (x963), 223-255. [7] P. G~mEL and M. Zis~a~, Calculus of Fractions and Homotopy Theory, Ergebnisse der Mathematik, vol. 34,

Springer, I967. [8] A. GROTHE~mXF.CK, S6minaire de g~om~trie alg~brique du Bois-Marie 196o/6i , SGA i, Lecture notes in mathe-

matics, n ~ 224, Springer, I97I. [9] I. M. HALL, The generalized Whitney sum, Quart. 07. Math. (2), 16 (I965) , 360-384 .

[xo] D. M. KAN, On c.s.s, complexes, Amer. 07. Math., 79 (1957) , 449-476. [1 I] J. W. MmNOR, On spaces having the homotopy type of a C-W complex, Trans. A.M.S., 90 (I959), 272-28o. [12] D. QUILLEN, The geometric realization of a Kan fibration is a Serre fibration, Proc. A.M.S., 19 (1968),

I499-I5oo. [I3] - - , Some remarks on etale homotopy theory and a conjecture of Adams, Topology, 7 (1968), 11 x-I x6. [I4] D. SULLIVAN, Geometric Topology, part I, M.I.T. Notes, x97o. [15] E. C. ZEEMAN, A proof of the comparison theorem for spectral sequences, Proc. Camb. Phil. Soc., 53 (1957),

57-62. [I6] J. P. MAY, SimOlidal objects in algebraic topology, D. Van Nostrand, 1967.

Manuscrit refu le 20 dgcembre 1970.

Rgvisg le 30 octobre 1971.

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