- Email: [email protected]
Crystalline conjecture via K-theory
Ann. scient. &. Norm. Sup., 4e strie, t. 31, 1998, p. 659 a 681.
CRYSTALLINE
CONJECTURE BY WIESEAWA
In Memoriam
VIA K-THEORY
NIZIOL
Robert Thomason
ABSTRACT. - We give a new proof of the Crystalline Conjecture of Fontaine comparing &ale and crystalline cohomologies of smooth projective varieties over p-adic fields in the good reduction case. Our proof is based on Thomason’s results relating &ale cohomology to algebraic K-theory 0 Elsevier, Paris
RBSUMB - Nous fournissons une nouvelle demonstration de la conjecture cristalline de Fontaine, qui compare les cohomologies &ale et cristalline de varietes projectives lisses sur les corps p-adiques ayant une bonne reduction. Notre preuve repose sur les resultats de Thomason reliant cohomologie ttale et K-theorie algbbrique. 0 Elsevier, Paris
1. Introduction The aim of this article is to show how the Crystalline Conjecture of Fontaine for projective schemes [8] (a theorem of Fontaine-Messing [l l] and Fakings [I?]) can be derived from Thomason’s comparison theorem [24] between algebraic and &ale K-theories. Let us recall the formulation of this conjecture. Let K be a co!mplete discrete valuation field of mixed characteristic (0, p) with ring of integers V and a perfect residue field. Let X be a smooth projective V-scheme. Assume that the relative dimension of X is less than p - 2 and that V is absolutely unramified. Then the conjecture postulates the existence of a natural period “almost-isomorphism” (a notion which can be made precise) a : H*(X,,
Z/‘pn) @ B,+, N H&(Xn/Vn)
@ B,+,,
where F is an algebraic closure of K, the subscript n denoting reduction mod p”, and where B,S, is a certain ring of periods introduced by Fontaine. The ring BL is equipped with a filtration, Galois action and a Frobenius operator, and the period “almost-isomorphism” is exlpected to preserve these structures. This, in particular, allows one to recover &ale cohomology from de Rham cohomology (with all the extra structures) and vice versa (Theorem 4.1). To prove the conjecture, by a standard argument (see, [ 111, [5]), it suffices to construct a ma.p a : H*(X,,
(1)
Z/p”)
+ H&(X&n)
@ B,+,
compatible with all the above structures, and, in addition, with Poincare duality and some cycle classes. ANNALES
SCIENTIFIQUES
DE L'BCOLE
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SUP~RIEURE.
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Our idea was to construct the map Q! via the higher K-theory groups. In the first step of the construction, one represents the right-hand side of (1) as a cohomology group of the scheme Xv, where v is the integral closure of V in K. Namely, using that B& n N Htr(V,/Vn, OV,,rJ 17, p.1051, we get that, by the Ktinneth formula, KxLlK)
@a
= Jm+Jr/n>
Hence, we need to construct a well-behaved &z : H”(-Q;
Z/P”(i))
~x,,,/vJ.
map
+ ql$$JV,,
6X~,JVn(i)),
at least for large enough i. Here, the twist in the crystalline cohomology refers to twisting the Hodge filtration and the Frobenius. Our construction is based on the following diagram F$‘F;+lKj(Xv; Z/p”) =-+ F;/F;+lKj(XK; Z/p”) j*
H,2r”-j(x-t;n/v,,c3x~,,,vn(i)) where l(j(.; Z/p”) The maps
=
H”“-qx~,
is me K-theory with coefficients and F$Kj(.; E$ :Kj(Xx;
Z/pn)
--+ H”“-qx~,
i$; :Kj(XT;
Z/p”)
--+ H,“,“-j(Xp,JVj,,
Z/p”(i)), Z/pm) is the y-filtration.
Z/p”(i)), c3x,,m,vn(i))
are the &ale [13] and the crystalline Chern class maps respectively. The arrow wij is to be constructed below. Our map azi-j,i will make this diagram commute. First, we prove (Lemma 3.1) that the restriction j* is an isomorphism. Next, using the work of Thomason and Soule, we show (Proposition 4.1) that, for large j and large p, the &ale Chern class map E$ is surjective and that the elements in its kernel are annihilated by a power of the Bott element ,& E Kz(XK; Z/p”). That allows us to construct the arrow wij in the above diagram: a map defined only modulo powers of the Bott element Pn, which makes the diagram commute. We set az;-j,i = 2:: wij. Then we prove that the Bott element maps via C$ to a non-zero divisor in the image of crystalline Chern class map. This gives that azi-j,; is a well-defined map. Our construction of the map c~-j,i makes it now easy to check its compatibility with Poincare duality and cycle classes. The above argument proves the Crystalline Conjecture for primes p larger than, roughly speaking, the cube of the relative dimension of the scheme X. This bound originates from the constant appearing in Thomason’s theorem. It is worse than the postulated bound (equal to the relative dimension plus 2). We also use this argument to prove the rational Crystalline Conjecture (Theorem 5.1). This time, we only need to assume that p # 2. This paper grew out of my collaboration with S. Bloch on a related project. It is a pleasure to thank him for many stimulating discussions, his help and his encouragement. W. Gajda helped me to learn &ale K-theory, C. Soul6 helped me to understand his paper 1211 and to simplify the proof of Proposition 4.1, B. Totaro read carefully parts of a preliminary version of this paper. I would like to thank them all for their help. Throughout the paper, let p be a fixed prime, let ??=denote a chosen algebraic closure of a field K, and, for a scheme X, let X, = X @ Z/p’“. 4e S!?RlE - TOME 31 - 19%
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2. Preliminaries Let V be a complete discrete valuation ring with its field of fractions K of characteristic 0 and with perfect residue field k of characteristic p. Let W(k) be the ring of Witt vectors with coefficients in k with its field of fractions Ka . Set GK = Gal(K/K), and let g be the albsolute Frobenius on IV(k). 2.1 The rings of periods. Consider the ring R = projlimv/pv, where V is the integral closure of V in K, the maps in the projective system being the p-th power maps. With addition and multiplication defined coordinatewise R, is a ring of characteristic p. Take its ring of Witt vectors W(R); then Bz is the p-adic completion of the divided power envelsope D,(W(R)) of the ideal &W(R) in W(R). Here < = [(p)] +p[(-1)], where (p), (-1) E R are reductions mod p of sequences of p-roots of p and - 1 respectively (if p # 2, we may and will choose (-1) = -1) and for J: E R, [XT]= [x,0,0, ...I E W(R) is its Teichmtiller representative. The ring B,f, is a topological W(k)-module having the following properties: 1. W(x) is embedded as a subring of BL and g extends naturally to a Frobenius 4 on B&; 2. Bz is equipped with a decreasing separated filtration FnBL such that, for n < p, $(FnB&) c pn B$ (in fact, PBZ is the closure of the n-th divided power of the PD ideal of Dr(IV(S))); 3. GK acts on BL; the action is W(k)-semilinear, continuous, commutes with 4 and preserves the filtration; 4. there exists an element t E FIB: such that 4(t) = pt, and GK acts on t via the cyclotomic character: if we fix E E R - a sequence of nontrivial p-roots of unity, then t = log([e]). B,, is defined as the ring BLb-“-, t-l] with the induced topology, filtration, Frobenius and the Galois action. For us, in this paper, it will be essential that the ring Bz can be thought of as a cohomology of an ‘arithmetic point’, namely [7, p.lOSj,
Let Bd+R= projlim(Q r
@ projlimBz,,/F’BL.n); Tl
BdR = Bd+R[t-I].
The ring Bd+Rhas a discrete valuat.ion given by powers of t. Its quotient field is BdR. We will denote by FnBdR the filtration induced on BAR by powers of t. 2.2. Crystalline representations. Assume that V = W(k). For the integral theory we will need the following abelian categories: 1. M.&(V) - an object is given by a p-torsion V-module n/r and a family of p-torsion V-modules FiM together with V-linear maps Fi M + F-lM, FiM + M and a-semilinear maps # : FiM -+ M satisfying certain compatibility conditions (see uw; 2. MF(V) - the full subcategory of MFbig(V) with objects finite V-modules M such that FiM = 0 for i > 0, the maps Fi(M) .+ M being injective and C Im & = M; 3. MJ$bl(V) - the full subcategory of objects M of MF(V) such that F”M = M and Fb+‘M = 0. ANNAILES SCIENTIFIQUES
DE L’lkOLE
NORMALE
SUPBRIEURE
662
w.
Consider the category M3[+ faithful1 functor L(M)
NIZIOL
(V) with b - a < p - 2. There exists an exact and fully
= ker(F’(M
@ SL(-b}(-b))
‘3
n/r @ Bz(-b)),
where f-b}, (-b) are the M3 and Tate twists respectively, from M31~,~1 (V) to finite Z,-Galois representations. Its essential image is called the category of crystalline representations of weight between a and b. This category is closed under taking tensor products and duals (assuming we stay in the admissible range of the filtration). We will need the following theorem. 2.1. - Let X be a smooth and proper scheme over V = W(k) of pure relative dimension d. For d 5 p - 2, H,“,(X/V, Oxlv/p”) lies in M~L~,,I(V). THEOREM
It was proved by Kato 217, 2.51 (the projective case), by Fontaine-Messing and by Faltings [5, 4.11. Here,
[II,
2.71,
is equipped with the Hodge filtration PH,“,(X/V,
OX,~/~“)
= Im(H”(X,,
$&J
+ H”(X,;
GJ~,)),
and the mappings 4” = 4*/p”
: F”H&(XlV,
Q&P”)
+ H,“,(X/V,
OX/V/P”),
where 4 denotes the crystalline Frobenius. 2.3. Syntomic regulators. Let X be a syntomic, that is, a flat and local complete intersection scheme over W(k). Recall the differential definition [17] of syntomic cohomology of Fontaine-Messing [ll]. Assume first that we have an immersion i : X c) 2 over W(k) such that 2 is a smooth W(k)- SCh eme endowed with a lifting of the Frobenius F : 2 --f 2. Let D, = Dx, (&) be the PD-envelope of X, in 2, and Jo, be the ideal of X, in D,. Consider the following complexes of sheaves on Xzar
where 4’ is the “divided by p”’ Frobenius. The complexes s,(T)x, sh(r)x are, up to canonical quasi-isomorphism, independent of the choice of i and F. In general, immersions as above exist locally, and one defines S,(T)X E D+(Xzar, Z/p”) by gluing the local complexes. Finally, one defines Safe E D+((X,),,,, Z/pm) as the inductive limit of s,(T)x”, , where V’ varies over the integral closures of V in all finite extensions of K in E. Similarly, we define sI(r)x and sk(r)x,. One sets Hi(X, 4e
Sl?RIE - TOME
S,(T)) := H&,(X, 31 - 1998 -
No
j
s,(T)~),
Hi(Xv,
G(T)> := H;,,(+,
sn(~)x~).
663
CRYSTALLINE CONJECTURE VIA K-THEORY
We list the following properties: (1) There is a long exact sequence
(2) There exists a well-behaved product
s&-)x RL S,(T’)X -+ Sn(T+ T/)x;
T + T’ L p - 1.
Since, for X smooth and proper over W(k) of relative dimension less than p, ~~,Fww),
Qx,/w(k)lPn) Z~XI;;IW~),
= Jfc,(wv4,
49
QX/W(k)/Pn)
= ~‘vcr(xI~(~),
@ B,+,,
~X/W(k)lP”)
[ll, 1.5,]
63 B,‘,)
Property (1) yields a natural map Hi(Xp,
&(r))
4 L(@,(X/W(k),
for
0x,w(lc,/p”){-r})
For a scheme X, let K,(X) ble the higher Quillen [ 191. Similarly, f or a noetherian scheme The corresponding groups, with coefficients Z/t and KL(X; Z/Z). Unless otherwise stated, we will K*(X; Z/Z) is an anticommutative ring. For a noetherian regular scheme X, set
p - 2 2 T > dimXK.
K.-theory groups of X as defined by X, let K:(X) be Quillen’s K/-theory. [24], will be denoted by Ki(X; Z/Z) assume that I # 2,3,4,8. In that case,
’ G(X) = i . (yi, (21) . . . yin (xn) I&(x1) = . . . = e(xn) = 0, il + . . . + i, > k, ) F,Kq(X; Z/p”) = (aq&~~) u . ’ . uyin(~,)Ja~~~Ko(X),io+il+...+i,
ifj 2 0, if j > 0,
2 k,),
where E is the augmentation on K,(X). We will use the following theorem of Gros [15]: THEOREM 2.2. - For any syntomic W(k)- SCh eme X and any integer 0 5 i < p - 1, there are functorial Chern classes
cy
: Kj (X) -+ H2+qX,
s&))
compatible with the crystalline Chem classes
Gros’ construction ffollows the method of Gillet [13], which can be extended easily [22] to yield functorial Chern classes ?Syn 2.7 : Kj(X;
Z/p”)
-+ @-j(X,
compatible with the classes c;sjy”. ANNALES
SCIENTIFIQUES
DE L’kOLE
NORMALE
SUPeRIELJRE
sn(i))
for j 2 2 and j = 0,
664
W. NIZIOL
LEMMA 2.1. - We record the following
1. 2. 3. 4.
properties of syntomic Chern classes: cZyn, for j > 0, is a group homomorphism. E%T, for j 2 2 and P # 2, is a group homomorphism. i$‘” are compatible with the reduction maps sn(i) -+ sm(i), n 2 an. Zf u: E Kl(X; Z/pn) and Q’ E K,(X; Z/p”), then c$n(cm’)=
(i - l)! - c E;~(a)c~;“(d): r+s=i (7. - l)!(s - l)!
assuming that 1; q > 2, 1 + q = j, 2i 2 j, i 1 0, p # 2. Moreover, if X is regular, then 5. If a E FJKo(X), j Z 0, and o! E FtKq(X; Z/p”), q 2 2, P # 2, then
E;$&Kd) = - (i(l:)~(kl)j)!c~~n(n)il;:n(a’). Similarly, for cJ?y&, if a’ E F,K,(X), k # 0. 6. The integral Chern class maps cam” restrict to zero on F;+‘Ko(X), i > 1. 7. The Chern class maps 2:;” restrict to zero on FG’lKj(X; Z/p”), j > 2, p # 2. Proof. - Recall the construction (due to Gillet [13]) of the higher characteristic classes E”“. The fastest way to describe it is as follows. One constructs universal classes Ci:” E Hzi(B.GL~/W(k),s,(i)) (de Rh am classes of the universal locally free sheaf on B.GLr,,/W,(k)). They yield compatible universal classes c$” E Hzi(X, GL&‘x), s&)), h ence a natural map of pointed simplicial sheaves on X, C;Yn : B.GL(Ox) -+ K(2i, S,(i),y), where K is the Dold-Puppe functor of r$&(i)~[2i] and s”lz(i)x is an injective resolution of sn(i)x. The characteristic classes i$‘” are now defined [13, 2.221 as the composition KJX;
Z/p”)
---f H -j(X,
Z x B.GL(&)+;
H-Yn)H-qX,
Ic(2i, 3&)x);
Z/p”) Z/p”)
-+ H-j(X,B.GL(Q)+; J-3 IF-qx,
Z/p”)
.s&)),
where B.GL(Ox) + is the (pointed) simplicial sheaf on X associated to the + -construction [21, 4.21. Here, for a (pointed) simplicial sheaf E. on X, H-j (X, 8.; Z/p”) = rj (qx, -2.); Z/P”) is the generalized sheaf cohomology of E. [13, 1.71: if we let ?$ denote the constant sheaf of j-dimensional mod p” Moore spaces, then H-j(X,E; Z/p”) = [T-J;; &.I, w h ere, for two pointed simplicial sheaves F,, F.’ on X, [F,, .F!] denotes the morphisms from F. to .F’ in the homotopy category. The map f is defined as the composition P(X,
Ic(2i, sxn(i)x); Z/p”) = 7q(K(S -S&)(X)>; Z/P">) 2 H,(K(2i, s”&)(X)); -+ Hj(CK(2i, S,(i)(X))) c Hj(NK(2i, S”&)(X))) = H&(i)(X)[2i]) = IF-qx, &(i))
Z/p”))
Here, hj is the Hurewicz morphism, and, for a simplicial abelian group A, CA and NA denote respectively the associated chain complex and the normalized chain complex. 4e
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Since, for p # 2, the Hurewicz morphism is compatible with addition and products [22, 11.2.21, so is the map f. The classes c:T are constructed in an entirely analogous manner. Since the map A,, : @ ---f ?$ A 7’5 is nullhomotopic for odd p, properties 1 and 2 follow as in [13, 5.261 from the Whitney sum formula for the syntomic classes of representations of sheaves of groups, which in turn follows from the fact that the de Rham cohomology of B.GL(n, m)/VV(k) is nontrivial only in even degrees (hence the even-degree syntomic cohomology injects into the de Rham cohomology) and the Whitney sum formula is true in the de Rham cohomology. Property 3 is easy to check. For the remaining properties, consider the following diagram PO B.GL(&)+ B.GL(&)+ A B.GL(CJx)+
Here, CSY” is the total Chern class, cl0 is the Loday multiplication, and * is the Grothendieck multiplication [2, 0.31. We claim that the above diagram commutes. This follows, as in [13, 2.321, from the tensor product formula for the syntomic classes of representations of sheaves of groups, which in turn follows, just like the Whitney sum formula above, from the fact that the de Rham cohomology of B. (GL,/W(k) x GL,/W(k)) is nontrivial only in even degrees. We have [20, 2.3.41 (C~;)*(~Yi)=C~l(alj...:~z,Yl;...;Yz), i>l
i>l
where ~(~~,...,~z,Y~,...,Yz)
120
=
(C,+,=z
G-~G-Y~
+
&(~)T,(Y))
and
G-.~
=
-(I
-
l)!/((r - l)!(s - l)!). H ere, 2, and 97, are polynomials of weight T and s respectively, and iE T + s = 1 then either 2, or T, is decomposable, i.e., can be represented as a sum of products of monomials of lower weight. To prove properties 4 and 5, it suffices to show that the terms &(z)T,(y) disappear when evaluated on the corresponding K-theory classes. For any q E Kj(X; Z/p”), j > 2, and any a, b 2 1, the following diagram commutes
-
B.GL(Ox)+
B.GL(Ox)+ A
ab
a6 1CW”ACW”
CSY” ,CW” 1
fq2(u
+ b), &(a + b)x)
Thus, since A,$ get Property 4.
is nullhomotopic,
ANNALES
DE L’lkOLE
SCIENTIFIQUES
NORMALE
=
A B.GL(Ox)+
q2a,
&(4x)
A lq2b, S,(b)x).
the decomposable terms 2, or T, disappear and we
SUP6RIEURE
466
W. NIZIOL
Assume now that we know that the Chern classes in properties 6 and 7 vanish on the monomials rk(z), k > i f 1. Notice that it suffices to prove properties 5, 6, and 7 for the products of monomials ye (z) : the general case will follow from the Whitney sum formula and property 2. We will thus assume now that all the K-theory classes are products of monomials as above. We will argue for property 5, the case of o’ E F$KO(X), k: f 0, and property 6 simultaneously, by induction on the number of monomials Ye in the appearing K-theory classes. If we use S/g (the simplicial version of the O-sphere) instead of F’i above, since (by induction from property 6) CzY”(a) = 0 for a < j (and similarly for a’), we get the vanishing of the terms Z,T, we wanted. Property 6 follows now by induction from the multiplication formula in property 5. We will argue for property 5, the case of cx’ E F$K,(X; Z/f), by the same type of induction. By induction from properties 6 and 7, all the terms Z,T, disappear. Property 7 follows by induction from the multiplication formula in property 5 and from property 6. It remains to show that 2:;” = 0 on elements of the form Yk (x), k > i + 1, II: E Kj(X; Z/p”) (and an analogous fact for the integral K-theory classes). We will reduce to a similar statement for the equivariant Chern classes of group representations. ecall how the y-operation Tk acts on Kj(X; Z/p”) [23, 41. Write K(GLN) for the Grothendieck group of representations of the group scheme GLN over Z. There is a morphism of abelian groups ,rAr : K(GLN)
-+ [Z x B.GLN(Ox)+,
Z x B.GL(Ox)+]
induced by a transformation, which to every representation p : GLN + GLM of the group scheme GLN associates a morphism of (pointed) simplicial sheaves B. GLN(13x) 4 B. G,?LM(0,). Passing first to the morphism induced on generalized cohomology groups, then to the limit over N, we get a morphism of abelian groups T : K(GL)
-+ Hom(Kj(X;
-+ Kj(X;
Z/p”)
Z/f)).
Now, the operation O/kis defined as the image under T of the element (Tk(IdN - N))N E K(GL), where Id,v, 1V are the classes of the natural, respectively trivial, representation of GLN. Prom the above description it is clear that it suffices to show that the composition B.GLnr(C?s)
3
I?.GL(Ox)+
2
Ic(2i, &&)x)
is homotopically trivial. Set, after Gillet [13, 2.271, N2i(X,
z x B.GL&Jx)+
, s&))
: = [Z x B.GLN(Ox)+,
ii*(X,
Z x B.GLN(Ox)+
, sn(*))
: = H”(X, n
Ic(2i, S”&y)];
Z x B.GLN(Ox)+, Hzi(X,
Z) x (1)x
Z x B.GLN(Ox)+,
sn(i)).
i>l
H*(X,Z
x B.GLN(O~)+,S,(*))
is a X-rmg [ 13, 2.271. Consider now the augmented
total Chern class map E : K(GLN) 4e
SGRIE - TOME
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31 - 1998 -
No
Z x B.GLN(&)+, 5
sn(x));
P ++ @4P)>
C.(P))
CRYSTALLINE
CONJECTURE
VIA
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K-THEORY
We claim that this map is a X-ring homomorphism. Indeed, it is clear that the map is well defined. Compatibility with addition and multiplication follows from the Whitney sum formula and the tensor product formula, which, as we have mentioned above, hold for the syntomic Chern classes. For the compatibihty with X-operations, in suffice_s to pass to the correspoding B.GL,M/W(IE) and check that c(/l\“Z) = A” C(E) E H*(B.GLM/W(~), sn(*)), where E is the universal locally free sheaf on B. GLM/W(~). As before, it suffices to check this formula in the de Rham cohomology of B. GLM/W(k). This can be done easily using the slplitting principle. Returning to the main line of the argument, we have to show that the class of cI~~(~yk(Id~ - N)) E H2i(X,GL~(Ox),s,(i)) is zero. This now is standard. Set XN = IdN -N. The above yields that the Chem polynomial cz:, is a X-ring homomorphism. Hence we have ci;ih(~N))
=
%(c$&V))
=
Yk(l+
c;;;((zN)t
= 1 + (-1)“~‘(k
+
C;y;(xN)t2
- l)!c;T;(xN)t’
+
. . .)
[2, 0.1.21
+ ...
Thus c~~~((r~(~~)) = 0 for 0 < i < Ic, as wanted. Similarly, by recalling [2, 0.1.21 again, we get that the integral Chern class maps cig”, i 2 1, are trivial on ~yk(5), k 2 i .-t 1, for any 2 E KO(.), E(X) = 0. Remark 1. - For any K-scheme
Y, one can paraphrase the a.bove proof to show that
the &ale Chern class’ maps
c$ : Kj(X) + H-(Y,
Z/p”(i)),
E$ : K,(X;
Z/p”)
+ Hy(Y;
Z/p”(i))
have analogous properties to those of the syntomic Chern classes. Similarly, working with the complex S;(T) instead of the complex S,(T), we get coholmology groups Hi (X, .sL(r)) and Hi (Xv, si (r)), which are targets of higher Chem classes [15, 2.31. All of the above listed properties of s,-Chem classes carry over to the s; -theory. For a V-scheme X, let Xa denote the special fiber of X. Recall that, for X smooth and proper over V, Kato-Messing [ 181 have constructed the following isomorphisms
H:,(Xo/W(k),Oxa,w(lc)) @Q @B,+, zr
Q @ projlimHk(Xi;i/W(k);
H&dWK)
k Bj--
E w$WQ
F’(H;,(&c/K)
@ pr0.in limH:,(Xv/W(k),
@B,+,)2
proj lim,(Q
We will need to know that ANNALES
[18, 1.2],
OX,~W(~)/P~)
SCIENTIFIQUES
DE L’!kOLE
NORMALE
SUPtjRIEURE
Ox~,~~Wn!k)lJ~N]) @ proilim
H:,(Xv/W(k),
[18, 1.41, $]/JiN])).
668
w. NIZIOL
LEMMA
2.2. - 1. The following
@ proj limHi(Xv; n
s;(r))
two compositions of maps are equal
+ proj lim,v(Q @ proj lim,H&(X~/W(k),
J~l/J~Nl))
where 6 is induced by the Berthelot-Ogus isomorphism (3, 2.21 Hz,(Xo/W(k)) @W(k) K E H&(&c/K). 2. For any smooth and projective scheme X over V, the Kato-Messing isomorphism her
:
Q@proj limHk&/W(k), 71
OXF/W(k)/pn)
=
Ox,,W(k))@Q@B~
H~,(&/W(lc),
is compatible on the image of the crystalline Chern classes 6:; : Ko(X)
-+ Q @ proj lim H,2,i(X/W(k), 72
ox,W@)/p”)
with the natural map
Q@
PM
lim,HZ,(XIW(~),
oX/W(k)/$)
Q@
-+
H:,(xO/w(@;
oXo,w(k)).
- Proof.- Letv = v @W(k) v, x = x @M(p) V, EC+,=
proj lim,Hzr(Vn,/V,). By the syntomic base change theorem [l, 2.3.51 H,*(V,/V,) N H&(7,/W,(k)) @w(k) V. Hence H&(pn/V,) = 0, for i > 0, and gz = J32 @W(k) V. In particular, 52 is flat over V. Using that and unwinding the definition of the Kato-Messing isomorphism h,,, we easily check that the following compositions of maps are equal
Q@
Q8
problimH~,(X~,n/W,(kj)
proj
% +
@:,(Xo/WV)
2
H:,(Xo,v/V)
limH~,(X~,TL/Wn(k))
H:,(x~/w(k)) @ w(k)
Q@
@v -+
Q@
proj
Q@
B:
@w(k)
Q@
@w(k)
V
BL
g,+,; lim
H$(Xv,n/W,(k))
n
f-
Q @ Hzr(X~/V)
-f+ H:,(Xo;v,‘V>
@V proj lim H,o,(%/V,)
72 @v Q @ @
Here, Xa,v = X0 @w(k) V, the map AZ is the base change morphism, and the map Xi is the Ktinneth morphism. This yields Property (1) of the lemma. Concerning Property (21, since the morphism
Xo: H:,(XoIWW @Q @B,t. -+ H:,(Xo/W(W @w(rc) Q @B; @w(k)V 4e
SERIE - TOME
31 - 1998 -
No
5
CRYSTALLINE
CONJECTURE
VIA
669
K-THEORY
is injcctive, it suffices to show that the composition
is the natural map H&(X1/W(k)) that tlhe composition
-+ H&(X,/V)
on the level of Chern classes, and
is compatible with the natural map H&(X/V) ---f Hz, (Xr /V) . The latter assertion is easy to show, and the former one follows from [3, 3.71.
3. K-theory
lemma
Let V be a complete discrete valuation ring with a field of fractions K. Let v denote the integral closure of V in F. LEMMA 3.1. - Let X be a smooth V-scheme. For any integer n, the natural morphism j* : Ki(X @V V; Z/n) + Ki(X @J,JK; Z/n) is an isomorphism. Praof. - We may assume the residue field of V to be separably closed. Let L/E/K be finite field extensions with rings of integers V, and If’, respectively. Let e be the ramification index of L/E. The map f : XV, -+ XV, being flat, the localization theorem in Ill-theory yields the following commutative diagram of maps of exact sequences
KI+,(XG Z/n)
-
Ki(X, @v, T/i,;Z/n)
-
q(X”,;
Z/n)
-
K;(xL; Z/n)
fg=,i* I T T Z/n)- q(XJ/,; Z/n) __fIC,'(XlJ; Z/n) q+,(xE; Z/n) - IqXK; flu,
* fx,
fXVET
k
The scheme Xz@vE I/L is non-reduced, with underlying reduced scheme XT, and we write i for the inclusion Xx c-+ Xx @vE V& We claim that fg- = ei,. Indeed, let M(Z) denote the exact category of coherent sheaves on a scheme 2” Then f-g- = f,” for the functor fK : M(X,) -+ M(Xzgv, VL), fK(F) = ;F@, VL. Since x@T:EkVLN $z]/se, filtering fK, we compute that f,” = CyzI J$, where every functor fz : M(Xz) -+ M(Xx@v, V’) is naturally equivalent to i,. From that, we easily see that there is a natural isomorphism inj lim K,((Xv,; LIK
Z/n) -% inj lim,,,Kl(XL;
z/n),
where the limit is over finite field extensions of K. Since the original scheme X was smoo’th, all the schemes XV, are regular, and we can pass to K-theory in the above isomorphism getting the statement of our lemma. ANNALES
SCIENTIFIQUES
DE L’lkOLE
NORMALE
SUP6RIEURE
W. NIZIOt
670
4. Integral
theory
Let the notation be that of section 2. Let X be a smooth quasi-projective V-scheme of relative dimension less than p - 2, V = W(k). We would like now to construct functorial Galois equivariant morphisms &zb
: H”(XE,
z/p”(b))
+
L(H,“,(x/v,
sX/P”){-b)).
We will be able to do it (4) only under certain additional restrictions on the integers a, b and d. Our construction will be based on the following diagram F@?;+lKj(XT;
Z/p”)
+
Z/p”)
F$‘F;+lKj(XK; 3
H”“-qxv;
s&))
H2i-qxz,
Z/p”(i)).
Here 1 5 i < p- 1, j 2 3, p” 2 5, p # 2. The Chern class map Cri” : F;Kj(Xv; Z/p”) + IS Hz”-j(X+&)) . d efi ne d as a limit over finite extensions V’/V of the syntomic Chern class maps F$ Kj (Xv) ; Z/p”) -+ Hzi-j(Xv,, s,(i)). Due to Lemma 2.1, the Chern class maps E$ and E:;” factor through FG’l yielding the maps in the above diagram. We will need to know how Bott elements in K-theory behave under higher Chern class maps. Recall that to every sequence of nontrivial p-roots of unity & E qPL 1. Namely, ,& is the unique element, which maps to 6 in the following exact sequence
o-+K~(Q~; z/p”)3 K,(Qp)c WQp)-+0. We will denote by ,& the unique element in K2 (&, ; Z/p”) mapping to ,& E K2 (Q,; Z/pn) (such an element exists and is unique by Lemma 3.1). We have a commutative diagram
d
K2(z,;
Z/P”)
-
K1(-Z,)
“,
where all the vertical maps are injections. Hence a(,&) LEMMA
$
= & as well.
4.1. - We have C”ly;(i,)
= t E H”(-z,, $:203n>
sn(l)),
L$$@n)
= CmE H’(Q,;
= t E Ho&,, Z/P”(~)),
where t E Bz( Z,) is the element associated to & *s (c$, section 2.1). Proof. - By SOL% [22, IV.1.31,
$2 : K2(Qp; Z/P") --f H’(Q,, 4e SBRIE - TOME 31 -
1998 - No 5
Z/P"(~))
s;(l)),
CRYSTALLINE
CONJECTURE
VIA
K-THEORY
671
is the Bockstein map followed by 1:he determinant map:
For the syntomic class, notice that we have an injection H”(zp, sn( 1)) ci ~,q.((-Z;,/P”wn(~), AL>. S’mce the syntomic classes and the crystalline classes are compatible, it suffices to show that Ed, = t E H~~((?&/p”)/VV~(k), Jn) N FlB,S,/p”. Consider the Kummer exact sequence n
0 + Z/p”(l)
+ G, % 6,
+ 0
on the Zariski site of Spec@,). Define classes of the natural representations ci(Idk) as the image via the boundary malp f%w(z,, GL@p), Z/V(l))
E
of the class of the GLk&,)- invertible sheaf associated to det (Idk). Paraphrasing the above mentioned argument of Soul6 one can show that, if we induce a map Cl,2
: K2(z,;
Z/P”>
-+
G,,(z,,
Z/P”(l))
via m’od p” Hurewicz morphism and via the Ktinneth homomorphism
from the classes cl(Idl,), then ~~,a(,&) = &. Consider now the following diagram
Kdzp; VP”)
where u : (Spec(&,/p”)/W,(k)),, -+ Spec(Z,)z,, is the projection from the crystalline to the Zariski topos and the map cv is defined by sending & to t on the open Spec(%,) and & to 0 on the open 62p. It suffices now to prove that this diagram commutes. Since both maps c1,2 and Eit2 are defined via the same procedure from characteristic classes of repres,entations (to see this for the crystalline classes, adapt the argument of Shekhtman [20, 2..2] to K-theory with coefficients), it suffices to prove that the map
maps cl(Idk)
to cy(Idk).
But the last classes come via Bockstein maps
G, 5 Z/p”(l)[-111, ANNALES
SCIENTIFIQUES
DE L’kOLE
NORMALE
G, SJPliRIELJRE
+ Gm,Z,pn l”~r~*&[-l]
672
w. NIZIOt
from the class of the GLI,(‘Z,)-’ mvertible sheaf associated to det(Idk). Here, d, 8”’ are the boundary maps coming from the Kummer sequence and from the exact sequence 0
+
(1 +u*J?Jzp)
-
(b*QJ(z,H*
il 0
-4
-+
i
0
il
II
+ (B&/p”)*
1 + FlB,S,/p”
(Z/pn)*
-+
Eh”)*
+
0,
respectively. Hence, it suffices to show that the following diagram commutes tin the derived category) d Gn %wN-11
It is clear on the open Spec(?&). For the commutativity the following commutative diagram with exact rows 0
+
-+
1+ FlBQp” T
0
i
on the open Spec(Z,),
(B&/p”)*
-+
+
--f
0
+
0
T
,7T
proj lim pPn(Zi)
(Z/pn)*
consider
proj lim Zi
--tp’
55;
where the maps in the projective limit are the p-th power maps, and the map y sends a sequence ( ui) to [(a;) 90, . . .] . This diagram yields the following commutative diagram
(z/p)*
5
(1+ F’B,s,/p”)[-l]
II zg
+
a
proj lim/+
F1%IP”I-K
(57;) [-l]
II
1 /App”(z;)[-l]
FlB,+,/p”[-11
II
T
21T
55; + a
J5
“,
J%+JP”[-11,
which completes our argument. For the s’-class, since the syntomic classes and the crystalline classes are compatible, it suffices to show that Cy:z(Dn) = t E H,q((zp/p”)/Wn(~),Jn) rv FlBL/p”. This was done above. In what follows, we will fix a generator < = (&) of Z,(l) and denote by ,&, jn, and t the elements corresponding to C. For a scheme Y over Z,, let &, E K2(Y; Z/p”) denote the pullback of the Bott element jn E Kz(z,; Z/p”). We will denote by Pn the image of pn in K~(YQ ; Z/p”). For a positive integer d, let N(d) denote the constant defined by Thomason inP[24]: N(d) = (2/3)d(d + l)(d + 2). 4" SI?RIE - TOME 31 - 1998 - No 5
673
CRYSTALLINE CONJECTURE VIA K-THEORY
PROPOSITION 4.1. - Let Y be a smooth quasi-projective scheme over -ii’. Let p # 2 and n be such that p” 2 5. Let i be any positive integer. There exists a constant T = T(d, i,j) depending only on the dimension d of Y and i, j such that, for j 2 2N(d), 1. the cokernel of~$ : FG/Fi+lKj(Y; Z/p”) + i!?j(Y, Z/p”(i)) is annihilated by T; 2. iv [x] is in the kernel of C$, then T[x] = [z] for z E FiKj(Y; Z/p”) annihilated by /3:(d). Moreover, for p > max{i + 1; (d t j + 3)/2}, we can take T = 1.
Proof. - The proposition follows from Thomason’s comparison theorem [24] between algebraic K-theory and Ctale K-theory, and from the degeneration of the Dwyer-Friedlander spectral sequence modulo torsion [21]. We get a uniform constant T as in the theorem through a careful bookkeeping of the appearing torsion - most of the work being done by Soul6 in [21]. Soul6 [21,4.1.4] has shown that tlhe algebraic Chern class c$ is equal to the composition of the topological Chern class cij : KF(Y; where Kp(Y; natural map
Z/p”)
Z/p”)
-+ IP-qy,
is the &ale K-theory
pj : K,(Y;
Z/p”)
from algebraic to &ale K-theory. Recall that we have a Dwyer-Friedlander E2S,4=
i > I, j > 2,
Z/p”(i)),
of Dwyer and Friedlander [4], [12], with the
+ KfyY;
Z/p”)
spectral sequence
H”(Y, Z/p”(i)) { 0
if 0 5 s 5 4 = 2i, otherwise
converging to Kp-,(Y; Z/p”), 4 - s > 3. Let F”KF(Y; Z/p”) denote the filtration on Kp(Y; Z/p”) defined by this spectral sequence. Consider also the following y-filtration F; = (y”(x)lk 2 i). S’mce the X-ring multiplication on KY (Y; Z/p”) is trivial, this version of y-filtration satisfies many of the formal properties of the filtration F;. In particular, we can prove, as in SoulC [21, 3.41, that M(d; i,j)F2i-jKF(Y;
(3)
Z/pn)
c F;KF(Y;
Z/p”)
c F2i-jKp(Y;
where the constant M(d, i, j) is to be defined below. NOW, we know [21, 4.21 that cij restricts to zero on Fzi-j+lKp(Y; inducles a map cij : F;/j?FlKF(y;
Zip”)
L!+ J$--j /F2i-j+2Ky(Y;
Z/p”),
Z/pn).
Z/pn) 3 l?-j(Y,
Hence, it
Z/p”(i)).
We will now study the maps f, y. Let’s first recall some constants from [21, 3.41. Let 1 be a positive integer, and let wl be the greatest common divisor of the set of integers kN (,@ - I), as k runs over the positive integers and N is large enough with respect to 1. Let M(k) be the product of the ~1’s for 21 < k. We will also need the integer M(k
ANNALES
m,
n)
=
r12m<21<2m--n+k+l
SCIENTIFIQUES
DE L’lkOLE
M(24.
NORMALE
SUPkRIEURE
474
W. NIZIOL
By the inclusions (31, the map f has kernel and cokernel respectively AI( d, i, j). Concerning the map g, notice that, by Soul6 [21, N2i-j(Y, Z/p”(i)) = Eii-“2i lies in the kernel K2i-j~2i T > 2, in the Dwyer-Friedlander spectral sequence. Hence, g : F~~-~/F2~-.i+2K~(yj
Zip”)
+ Kzi-izi
annihilated by M(d, i + 1, J’), 4.21, the image of cij in of all higher differentials d,, we have a factorization
L+ Hz”-j(y,
Zip”(i)).
Since [21, 3.3.21 M(d)& = 0 for any T > 2, the cokernel of the inclusion K2i--ja2i q H2i-j(Y, Z/p”(i)) is annihilated by M(d). Consider now the composition E2i-j,2i cc
,$-j/j@-j+2Kft(y; h
J$i+,F%j+2Kj”t(y;
Zip”)
i Zip”)
pi2i-j,2i =
$72~‘j,2i,
where $;j is the natural projection. This composition is proved in [21, 4.21 to be equal to multiplication by (-l)i-l(i - l)!. H ence the kernel of g is annihilated by (i - 1) !. Also, since &f(d)&. = 0 for any T > 2, the kernel and cokernel of & are annihilated by n/i(d). Hence the cokernel of g is annihilated by M( d)2 (i - 1) !. The above yields that the kernel of Cij is annihilated by Ill(d, i + 1; j)(i - I)! and its cokernel by &I(d;i,j)IU(d)‘(i - l)!. Consider now the map
We know from,Thomason [24], [25, 3.21, that ,5j is-surjective for j > 2N(d). For its kernel, let 5 E FGK,(Y; Z/p”) be such that pj(z) E F$+lKp(Y; Z/p”). If j > 2N(d), there exists a y E F;+lKj(Y;Z,p’“) such that pj(z) = pj(y), or pj(z - y) = 0. Hence, if we set z = IC - y, then y = z - z E F;+‘Kj(Y; Z/p”) and z is annihilated by @r(d) (again by Thomason). Combining all of the above, we get that, for j 2 2N(d), the cokernel of E$ is annihilated by M(d, i,j)M(d)‘(i - l)!, and its kernel, restricted to F;Kj(Y; Z/p”)/@G+lKj(Y; Z/p”), is, modulo M(d, i + l,j)(i - l)!, annihilated by ,&fcd). Since M(2i)F;Kj( Y; Z/p”) c FGKj(Y; Z/p”) + F:+lKj(Y; Z/p”) (argue as in [21, 3.4]), we can take T = T(d, i,j) = (i - l)!n/i(2i)AJ(d, i,j)AJ(d, i + 1:j)M(d)2. Since an odd prime p divides &I(d, i, j) if and only if p < (J’ + d + 3)/2, and divides 0 M(Z) if and only if p < (l/2) + 1, we get the last statement of the proposition. Remark 2. - The results of Soul6 needed in the above proof are stated in [21] for the p-adic topological K-theory. Working with Friedlander’s definition of &ale K-theory mod pn 1121, i.e., using systematically P”(p”) A Xkt, where P”(p”) is the simplicial Moore space obtained from the mapping cone of multiplication by p” on the circle and where, for a scheme X, X6, is its &ale topological type, instead of Xbt, one sees that Soule’s results hold as well for the p-torsion K-theory (with the usual restriction on the degree of the K-groups) and his proofs carry over to that case almost verbatim. Let now p, a, b, d satisfy the following condition
(4 4e
b 2 d, p - 2 2 max{b + N(d), (1/2)(d S&RIE - TOME
31 - 1998 -
No
5
+ 2b - a - l)},
2b - a 2 2N(d).
CRYSTALLINE
CONJECTURE
VIA
675
K-THEORY
Define the morphisms %b
: Ha(Xz,
z/p”(b))
-+
L(H,“,(x/v,
oX,v/~“){-b~)
as the composition oab := ~,C~~bb--a(j*)-l(C~~~b-a)-l, where &, is the natural map H”(XF, s,(b)) * L(ffFr(X/V, CJ&v/p’“){--b}). Here (2””b,a&a)-l (x) is defined by taking any element in the preimage of x. We do not know whether this is well-defined, but we do know that its composition with q!~~~~~~-~(j*)-’ is. Indeed, by the above proposition, an ambiguity in the definition of (el;b-a)-l may only come from a class of z E E~=&-a(X~; Z/p") such that pzCd)z = 0. So bt’“‘(j*)-l(z) = 0. Hence
’ = ~~[~~~(d),zb+2N(d)-~~~~~(d)(j*)-1(r))l = (-l)N(d)(N(d)
+ b - l)!/(b
- l)!~n[~~~(~~(d))~~~~b-~((j*)-l(~))]
= (-l)N(d)(N(d)
+ b - l)!/(b
- l)!~“‘“‘?i,l~~~~b-~((~,~)-l(~)).
To conclude that $~~Ci~t~~,((j*)-~ I(z)) = 0, as wanted, it suffices now to show that, for b + 1 < p - 2, mult$ication by t
is an isomorphism.
But the multiplication
map
induces a map L(1Z&(X/V,
ox/v/P”){-b})
8 L(V/P”{-1))
+ L(JJ,“,(XIV,
which, since H$(X/V, 0 x,v/p”){--b - l} E MFL-b-i,a](V) isomorphism. Since multiplication by t yields an isomorphism [9, 5.3.61, hence an isomorphism L(FxX/V
-4, L(q$qV
Qx/vlP”){-w
Ox/v/P”){-b})
CC/v/P”){-b
- I>)>
(Theorem 2.1), is an Z/pn 2 L(V/p”{ - 1))
@ L(V/p”{-l}),
we are done. Assume now that X is projective of pure dimension cl. We will show that in that case the maps o&b are isomorphisms. First we prove 4.2. - 1. The maps o&b commute with products. 2. The maps c&b are compatible with some cycle class maps, i.e., if Z is an irreducible closed subscheme of codimension j in X, smooth over V, then
LEMMA
aQ,b(d”t(zK)[b-‘) = d=‘(Z~,)t~-~~. Here clCr(Z,) E H,“,j (X,/V) ANNAL.ES SCIENTIFIQUES
DE L’kOLE
NORMALE
is the one dejined by Gros in [ 161. SUPkRIEURE
676
W. NIZIOL
3. For irreducible X5
the following H2”(X,,
diagram commutes
Z/p”(b))
q+
ZlPV
piI
4d,b I
- 4
Proof. - Property 1 follows from the multiplication formula in Lemma 2.1.5. For property 2, let [0,] and [(3zK] denote the class of 0~ and OZK in Ka(X) and K~(XK) respectively. From the hypotheses, both 2~ and .Zk are integral and of codimension j in their respective fibers. We will need to do some preliminary computations. Let s(d) = fli?$: AY!.First, we claim that s(d)[0,] E F$Ka(X) (it follows then that s(d)[0,,] E F$Iyo(XK)). By the Integral Riemann-Roth [23, 4.61, for any z E F$&(Z) T;~($+(.z)) = i,((-l)‘+“-l(~ + 1 - l)!z + y), where i, : Ka(Z) + Ka(X) and ‘+lKo(Z). Since F ‘+‘K O(2) = 0, this gives our claim. NeXT, we claim that, for 0 < i < j, the Chern classes c$a([OzK]) and c~~([U3,]) are zero. For the Ctale Chern class, since for an open U in XK such that the codimension of XK \ U is strictly greater than i, the restriction H~~(XK, Z/p”) -+ Hij( U, Z/p”) is an injection, we may assume that ZK is the zero-scheme of a regular section s of a locally free sheaf E of rank j on XK. The Koszul complex gives a resolution
Hence [0,,] = C(--l)‘[Ai(c)] in Ka(Xh’). 0 ur claim now follows from [2, 0.1.21. For the syntomic Chern class, notice that $~~cfridl([13,]) = c~~~([C?~,]). Recall that the crystalline Chern classes factor through the logarithmic de Rham-Witt classes ~~~lfo([~Z,I) E @(XI,, wLf&k,log ). Hence it suffices to show that C;:a(IO,,]) = 0. This can be shown exactly as in the &ale case (for the injection of the restriction map @(Xk: wnqyk,log ) --+ Hi(U, Wnfl&,lOg) refer to the purity result in the logarithmic de Rham-Witt cohomology [ 16, 11.4.1. I]). Finally, from the Whitney sum formula we have
= 4~)~ncg~;;,^([~z]). ~;~owwZKl)= 4~)c~~il([~Z,l), ?k;~~b”(~(~)~~z1) We are now ready to prove property 2. Since c$@,) = & (& corresponding to &) we get from the multiplication formula in and c$([OZK]) = (-l)j-“(j - l)!clbt(ZK), Lemma 2.1.5, we get that --Bt C6,2(b-j)(S(d)(C?ZK]P~-j)
= (-l)bqJ
- l)!s(d)cl”t(&)&j.
On the other hand, ?i/~i$~~(&) = t and &cg~~([Sz]) IV.3.1.21. Hence 1cl,li~~~~6-I)(s(d)[~z;~~-i)
= (-l)b-l(b
- TOME 31 -
1998 - No 5
[16,
- l)!s(cqclc’(z,)tb-~.
By the definition of the map d!aj,b, since our p > max{d+j 4e &RIE
= cjcfO([OZk:]) = clc’(ZI,)
- 1, b- 11, this gives property 2.
CRYSTALLINE
CONJECTURE
VIA
677
K-THEORY
Concerning property 3, we can assume that the residue field of V is algebraically closed. Let f’ be a rational point of X over V. Recall that tret and trC’ are characterized by the mapping ~1”~(PK ) C-“, respectively cl”‘(E’k), to 1. It suffices thus to show that Q;d,b(cl”t(P&-d)
= clcr(P#-d,
which is a special case of property 2.
0
4.1. - For any projective, smooth V-scheme X of pure relative dimension d, the Calois equivariant morphism THEOREM
is an isomorphism, if the numbers p, b, d sati&
b 1 2d + N(2d), p - 2 > 2b + N(2d).
Proof. - We may assume that k is algebraically closed and that X is irreducible. The line of the argument is standard [ll], [6, 2.41. Both the target and the domain satisfy Poincare duality:
Ha&,
Z/p”(b)) 63H”“-‘“(X,, 2 H2d(Xz,
wf,“,(wK
Z/p”(2b))
Qc/vlP”)W))
-5 L(H;f(x/v,
Z/p”(b)) tz Z/pn(2b - d),
(23WL, ““-“(x/v,
0 x,v/p”){-2b})
~x/v/P”){-b})
5 b(V{-2b
+ d}) z Z/p”(2b
- d)
By L,emma 4.2, c&b has a left inverse 0,-d-. To show that GE;: is a right inverse as well, it would suffice to show that it commutes with products. Since it commutes with Ktinneth products (because CY,~is functorial and commutes with cup products), it suffices to show that it commutes with A*, A : X c+ X x X being the diagonal embedding, or equivalently, that c&b commutes with A,. That, in turn, would follow, if we would show that o,b respects class of the diagonal and that A* is surjective. But this holds, since piA = Id, where p1 : X x X + X is the projection onto the first factor, and we have Lemma 4.2. 0 Remark 3. - The lower bound on the prime p appearing here is of order d3, while Faltings in [5] gets an order of d. While it is perhaps possible to decrease our lower bound slightly by a more careful argument, the main obstruction comes from the constant N (d:) of Thomason. It seems reasonnable to believe that the optimal N(d) should be of order d as well.
5. Rational
theory
Throughout this section, let the prime p be odd. Let X be a smooth quasi-projective V-scheme, where now the ring V is possibly ramified over W(k). For large b and p # 2, we will construct Galois equivariant functorial morphisms
ANNALES
SCIENTIFIQUES
DE L’lkOLE
NORMALE
SUPliRlEURE
678
W. NIZIOL
Most of the work was already done in section 4. For j > 3, p” > 5, p # 2, since Lemma 2.1 holds in this context as well, we get the following diagram F;/F;+l
For b > d + N(d),
Kj (X,; Z/p”)
+
3
F;/F;“Kj(Xi7;
Z/pn)
define the morphisms
as the composition
where $J~ is the natural projection
and we set I(b; d) = (-i)“(d)(N(d) + b - l)!/(b - I)!. Here (C&-a)-l(~) is defined by taking any element in the preimage of T(d; b, 2b - a)z (by Proposition 4.1, T(d, b, 2b - a)~ lies in the image of T$&-~ >. Again, by Proposition 4.1, any ambiguity in that definition comes from a class of y such that T(d, b, 2b - a)[~] = [z] and z E Kzbpa(XK; Z/p”) is annihilated by /3zCd). Hence
= ~n[~~~(d):2b+2N(d)--a(~~(d)(j*)-1[Zl)1
= o
and the map atb is well defined. Let now p # 2 and b 2 d + N(d). Define the morphism
p(b))-+ fCXfolW(O Qxo/w(q) @w(k)Rx{-b} as the composition of Q @ proj lim,aEb with the Kato-Messing isomorphism @pi = H,n,(Xokk),
iim~~(X~/W(IC>,O~,/w(k)/p~) Qx,,w(~))
@w(lc) K'i
@Q
and the division by Z(b, d)T(d, b, 2b - a)‘t”cd). THEOREM 5.1. - Let p # 2 and let X be any projective smooth V-scheme of pure relative dimension d. Then, assuming b 2 2d + N(2d), the morphism
is an isomorphism. 4e S~RRIE- TOME 31 - 1998 - NO 5
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Moreover, the map Qi,b preserves the Frobenius and the action of Gal(K/K), extension to BdR, induces an isomorphism of @rations. Proof. -- We have the following
analogue of Lemma 4.2.
LEMMA 5.1. - 1. Let z E H”(Xx:, i(b, d)Z(e, d)K(b,
and, after
Z/p”(b))
and y E H”(XF,
e)t N(d)T(d, b, 2b - c~)~T(d,
= l(b + e; d)K(b,
e)T(d,
Z/p”(e)).
e; 2e - c)~cc~+~,~+~(z
Then U y)
b + e; 2b + 2e - u - c)2azb(x)
u aze(y),
where K(b,e) = (b + e - l)!/(b -- l)!(e - l)!, assuming that all the indices are in the valid range. 2. The maps C&b are compatible with some cycle class maps, i.e., if Z is an irreducible closed subscheme of codimension j in X, smooth over V, and b 2 j, then n2&1”“(Z#-9
= clc’(Z,,)tb+.
Here cl”‘(Zk) E H,“,j(Xo/V) is the one defined by Gros in [16]. 3. For b 2 d and irreducible X;k, the following diagram commutes
Q&W
H2d(X~T,
-
tP
Q&b - 4 p--d
~Zd.26 1 H,2,d(&/WJc),
Qx,,w(&-2b)
1 @ Bcr
trc’
Bcr{d - 2%
Proof. - For properties (1) and (2) we evoke Lemma 4.2 and argue as in the proof of Lemma 2.2.2. For property (3), take a rational point P of X over V. Since t@ and trCr are characterized by the mapping ~1”~(PK)<~~-~, respectively clcr(Pk), to 1, it suffices to show that
Q2d,2b(Cl~e(PK)C2b-d) = ,~c’(P,)t2b-d. But that follows from property (2). 0 Assume that k is algebraically closed and that X is irreducible. By construction o!,b is a Galois equivariant map compatible with the Frobenius action. Since both sides have the same rank over B,,, it suffices to show that the morphism (u,b is injective. This follows from the fact that both the domain and the target satisfy Poincare duality, and the map o,b commutes with products and traces (by the above lemma). The proof that c&b induces an isomorphism on filtrations is now standard (see, [18, 31). By Lemma 2.2.1 extension of d!,b to BdR is compatible with the filtrations. One passes to the associated grading and reduces to showing that the induced map s’ab : c,(z) Q,
jy(XF,
Q,(b))
-4 $ C,(b + ljCZ
j) 8.~ H”-‘(X,,
$ii,,)l
1 E ‘,
is an isomorphism (this is the statement of the Hodge-Tate Conjecture). The dimensions of both sides being equal, it suffices toI show that i&b is injective. Since both the target and the domain of ?&, satisfy Poincare duality and &b is compatible withi products (Lemma 5.1. I), ANNALES
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for i&b to have a left inverse, it is sufficient that it be compatible with traces. But, &d,ab is obtained from the composition of the map
a;d,2b: H2d(X,,
,(2b)) -+ Q @ proj lim,H,$(Xv/W(k),
JFbl/JFb+ll)
induced by grading the maps o&2b, and the natural isomorphism ggr : Q @ proj Sm,H,2,d(Xv/W(k),
Ji2b1/JFb+11) N 6,(26
118, 1.41
- d) @ Hd(XK,
fl$K,zv).
Since the Hodge trace is compatible with the de Rham trace, we can use Lemma 2.2.1, Lemma 5.1.3, and the compatibility of the Berthelot-Qgus isomorphism with traces [18, B.3.31 to conclude our H:,(Xo/W(~)) @w(rc) K - H&(X,/W proof. 0
REFERENCES [I]
Theorie de Dieudonne’ cristalline. II, Lect. Notes in Math., vol. 930, York, 1982. P. BERTHELOT, A. GROTHENDIECK and L. ILLUSIE, Theorie des intersections et theoreme de Riemann-Roth, Lect. Notes in Math., vol. 225, Springer-Verlag, Berlin, Heidelberg and New York, 1971. P. BERTHELOT and A. OGUS, F-isoctystals and de Rham cohomology. I, Inv. Math. 72, 1983, pp. 159-199. W. DWYER and E. FRIEDLANDER, Algebraic and &ale K-theory, Trans. Amer. Math. Sot. 292, 1985, no. 1, pp. 247-280. G. FALTINGS, Crystalline cohomology and p-adic Galois representations, Algebraic analysis, geometry and number theory (J. I. Igusa ed.), Johns Hopkins University Press, Baltimore, 1989, pp. 25-80. 6. FALTINGS, p-adic Hodge-theory, J. of the AMS 1, 1988, pp. 255-299. J.-M. FONTAINE, Cohomologie de de Rham, cohomologie crystalline et representations p-adiques, Algebraic Geometry Tokyo-Kyoto, Lect. Notes Math., vol. 1016, Springer-Verlag, Berlin, Heidelberg and New York, 1983, pp. 86-108. J.-M. FONTAINE, Sur certains types de representations p-adiques du groupe de Galois d’un corps local, construction d’un anneau de Barsotti-Tate, Ann. of Math. 115, 1982, pp. 529-577. J.-M. FONTAINE, Le corps des periodes p-adiques, A&risque, vol. 223, 1994, pp. 321-347. J.-M. FONTAINE and G. LAFAILLE, Construction de representations p-adiques, Ann. Sci. .Ec. Norm. Sup., IV. Ser., 15, 1982, pp. 547-608. J.-M. FONTAINE and W. MESSING, p-adic periods and p-adic &ale cohomology, Current Trends in Arithmetical Algebraic Geometry (K. Ribet, ed.), Contemporary Math., vol. 67, Amer. Math. Sot., Providence, 1987, pp. 179-207. E. FRIEDLANDER, l&ale K-theory. II. Connections with algebraic K-theory, Ann. Sci. Ecole Norm. Sup., IV. Ser., 15, 1982, pp. 231-256. H. GILLET, Riemann-Roth theorems for higher algebraic K-theory, Adv. Math. 40, 1981, pp. 203-289. H. GILLET and W. MESSING, Cycle classes and Riemann-Rock for crystalline cokomology, Duke Math. J., 55, 1987, pp. 501-538. M. GROS, Regulateurs syntomiques et valeurs defonctions Lp-adiques I, Invent. Math., 99, 1990, pp. 293-320. M. GROS, Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmic, MCm. Sot. Math. France (N.S.) 21, 1985. BZ. KATO, On p-adic vanishing cycles (application of ideas of Fontaine-Messing), Algebraic Geometry, Sendai, 1985, Adv. Stud. Pure Math. 10, North-Holland, Amsterdam-New York, 1987, pp. 207-251. K. KATO and W. MESSING, Syntomic cohomology and p-adic &tale cohomology, TBhoku Math. J. 44, 1992, pp. 1-9. D. QUILLEN, Higher algebraic K-theory I, Algebraic K-theory I, Lecture Notes in Math. 341, Springer-Verlag, Berlin-Heidelberg-New-York, 1973, pp. 85-147. V. V. SHEKHTMAN, Ckern classes in algebraic K-theory, Trans. Moscow Math. Sot., Issue I, 1984, pp. 243-271. P. BERTHELOT,
L. BREEN and W. MESSING,
Springer-Verlag,
[2] ;3] [4] [5] [6] [7]
[8] [9]
[lo] [I I]
[ 121 [13]
[14] [ 151 [16] [17] [18] [19] [ZO]
4e SeRIE - TOME 31 -
Berlin-New
1998 - No 5
CRYSTALLINE [21] C. [22] C. [23] C. [24] R. [25] R.
CONJECTURE
VIA
SOULS, Operations on &ale K-theory. Applications., Algebraic K-theory I, Lect. Notes Math. 966, Springer-Verlag, Berlin, Heidelberg and New York, 1982, pp. 271-303. SOUL& K-thkoorie des anneaux d’enti,ers de corps de nombres et cohomologie &ale, Inv. Math. 55, 1979, pp. 251-295. SOUL& Ope’rations en K-thkorie algibbrique, Canad. .I. Math. 37, no. 3, 1985, pp. 488-550. THOMASON, Algebraic K-theory and &ale cohomology, Ann. Scient. Ecole Norm. Sup. 18, 1985, pp. 431-552. THOMASON, Bott stability in algebraic K-theory, Applications of algebraic K-theory to algebraic geometry and number theory I, II (Boulder, Colo., 1983), Contemporary Math., vol. 55, Amer. Math. Sot., Providence, 1986, pp. 389-406.
(Manuscript received accepted after revision
W. N~z~ot Department of Mathematics, College of Science, University of Utah, Salt Lake Utah 84112-0090, USA SE-mail:
ANNALES
681
K-THEORY
City
[email protected]
SCIENTF'IQUES
DEL'kOLENORMALE
SUPrjRIEURE
October March
21, 1996; 20, 1998.)
CRYSTALLINE
CONJECTURE BY WIESEAWA
In Memoriam
VIA K-THEORY
NIZIOL
Robert Thomason
ABSTRACT. - We give a new proof of the Crystalline Conjecture of Fontaine comparing &ale and crystalline cohomologies of smooth projective varieties over p-adic fields in the good reduction case. Our proof is based on Thomason’s results relating &ale cohomology to algebraic K-theory 0 Elsevier, Paris
RBSUMB - Nous fournissons une nouvelle demonstration de la conjecture cristalline de Fontaine, qui compare les cohomologies &ale et cristalline de varietes projectives lisses sur les corps p-adiques ayant une bonne reduction. Notre preuve repose sur les resultats de Thomason reliant cohomologie ttale et K-theorie algbbrique. 0 Elsevier, Paris
1. Introduction The aim of this article is to show how the Crystalline Conjecture of Fontaine for projective schemes [8] (a theorem of Fontaine-Messing [l l] and Fakings [I?]) can be derived from Thomason’s comparison theorem [24] between algebraic and &ale K-theories. Let us recall the formulation of this conjecture. Let K be a co!mplete discrete valuation field of mixed characteristic (0, p) with ring of integers V and a perfect residue field. Let X be a smooth projective V-scheme. Assume that the relative dimension of X is less than p - 2 and that V is absolutely unramified. Then the conjecture postulates the existence of a natural period “almost-isomorphism” (a notion which can be made precise) a : H*(X,,
Z/‘pn) @ B,+, N H&(Xn/Vn)
@ B,+,,
where F is an algebraic closure of K, the subscript n denoting reduction mod p”, and where B,S, is a certain ring of periods introduced by Fontaine. The ring BL is equipped with a filtration, Galois action and a Frobenius operator, and the period “almost-isomorphism” is exlpected to preserve these structures. This, in particular, allows one to recover &ale cohomology from de Rham cohomology (with all the extra structures) and vice versa (Theorem 4.1). To prove the conjecture, by a standard argument (see, [ 111, [5]), it suffices to construct a ma.p a : H*(X,,
(1)
Z/p”)
+ H&(X&n)
@ B,+,
compatible with all the above structures, and, in addition, with Poincare duality and some cycle classes. ANNALES
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- 0012-9593/98/05/O
Elsevier, Paris
660
W. NIZIOE
Our idea was to construct the map Q! via the higher K-theory groups. In the first step of the construction, one represents the right-hand side of (1) as a cohomology group of the scheme Xv, where v is the integral closure of V in K. Namely, using that B& n N Htr(V,/Vn, OV,,rJ 17, p.1051, we get that, by the Ktinneth formula, KxLlK)
@a
= Jm+Jr/n>
Hence, we need to construct a well-behaved &z : H”(-Q;
Z/P”(i))
~x,,,/vJ.
map
+ ql$$JV,,
6X~,JVn(i)),
at least for large enough i. Here, the twist in the crystalline cohomology refers to twisting the Hodge filtration and the Frobenius. Our construction is based on the following diagram F$‘F;+lKj(Xv; Z/p”) =-+ F;/F;+lKj(XK; Z/p”) j*
H,2r”-j(x-t;n/v,,c3x~,,,vn(i)) where l(j(.; Z/p”) The maps
=
H”“-qx~,
is me K-theory with coefficients and F$Kj(.; E$ :Kj(Xx;
Z/pn)
--+ H”“-qx~,
i$; :Kj(XT;
Z/p”)
--+ H,“,“-j(Xp,JVj,,
Z/p”(i)), Z/pm) is the y-filtration.
Z/p”(i)), c3x,,m,vn(i))
are the &ale [13] and the crystalline Chern class maps respectively. The arrow wij is to be constructed below. Our map azi-j,i will make this diagram commute. First, we prove (Lemma 3.1) that the restriction j* is an isomorphism. Next, using the work of Thomason and Soule, we show (Proposition 4.1) that, for large j and large p, the &ale Chern class map E$ is surjective and that the elements in its kernel are annihilated by a power of the Bott element ,& E Kz(XK; Z/p”). That allows us to construct the arrow wij in the above diagram: a map defined only modulo powers of the Bott element Pn, which makes the diagram commute. We set az;-j,i = 2:: wij. Then we prove that the Bott element maps via C$ to a non-zero divisor in the image of crystalline Chern class map. This gives that azi-j,; is a well-defined map. Our construction of the map c~-j,i makes it now easy to check its compatibility with Poincare duality and cycle classes. The above argument proves the Crystalline Conjecture for primes p larger than, roughly speaking, the cube of the relative dimension of the scheme X. This bound originates from the constant appearing in Thomason’s theorem. It is worse than the postulated bound (equal to the relative dimension plus 2). We also use this argument to prove the rational Crystalline Conjecture (Theorem 5.1). This time, we only need to assume that p # 2. This paper grew out of my collaboration with S. Bloch on a related project. It is a pleasure to thank him for many stimulating discussions, his help and his encouragement. W. Gajda helped me to learn &ale K-theory, C. Soul6 helped me to understand his paper 1211 and to simplify the proof of Proposition 4.1, B. Totaro read carefully parts of a preliminary version of this paper. I would like to thank them all for their help. Throughout the paper, let p be a fixed prime, let ??=denote a chosen algebraic closure of a field K, and, for a scheme X, let X, = X @ Z/p’“. 4e S!?RlE - TOME 31 - 19%
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2. Preliminaries Let V be a complete discrete valuation ring with its field of fractions K of characteristic 0 and with perfect residue field k of characteristic p. Let W(k) be the ring of Witt vectors with coefficients in k with its field of fractions Ka . Set GK = Gal(K/K), and let g be the albsolute Frobenius on IV(k). 2.1 The rings of periods. Consider the ring R = projlimv/pv, where V is the integral closure of V in K, the maps in the projective system being the p-th power maps. With addition and multiplication defined coordinatewise R, is a ring of characteristic p. Take its ring of Witt vectors W(R); then Bz is the p-adic completion of the divided power envelsope D,(W(R)) of the ideal &W(R) in W(R). Here < = [(p)] +p[(-1)], where (p), (-1) E R are reductions mod p of sequences of p-roots of p and - 1 respectively (if p # 2, we may and will choose (-1) = -1) and for J: E R, [XT]= [x,0,0, ...I E W(R) is its Teichmtiller representative. The ring B,f, is a topological W(k)-module having the following properties: 1. W(x) is embedded as a subring of BL and g extends naturally to a Frobenius 4 on B&; 2. Bz is equipped with a decreasing separated filtration FnBL such that, for n < p, $(FnB&) c pn B$ (in fact, PBZ is the closure of the n-th divided power of the PD ideal of Dr(IV(S))); 3. GK acts on BL; the action is W(k)-semilinear, continuous, commutes with 4 and preserves the filtration; 4. there exists an element t E FIB: such that 4(t) = pt, and GK acts on t via the cyclotomic character: if we fix E E R - a sequence of nontrivial p-roots of unity, then t = log([e]). B,, is defined as the ring BLb-“-, t-l] with the induced topology, filtration, Frobenius and the Galois action. For us, in this paper, it will be essential that the ring Bz can be thought of as a cohomology of an ‘arithmetic point’, namely [7, p.lOSj,
Let Bd+R= projlim(Q r
@ projlimBz,,/F’BL.n); Tl
BdR = Bd+R[t-I].
The ring Bd+Rhas a discrete valuat.ion given by powers of t. Its quotient field is BdR. We will denote by FnBdR the filtration induced on BAR by powers of t. 2.2. Crystalline representations. Assume that V = W(k). For the integral theory we will need the following abelian categories: 1. M.&(V) - an object is given by a p-torsion V-module n/r and a family of p-torsion V-modules FiM together with V-linear maps Fi M + F-lM, FiM + M and a-semilinear maps # : FiM -+ M satisfying certain compatibility conditions (see uw; 2. MF(V) - the full subcategory of MFbig(V) with objects finite V-modules M such that FiM = 0 for i > 0, the maps Fi(M) .+ M being injective and C Im & = M; 3. MJ$bl(V) - the full subcategory of objects M of MF(V) such that F”M = M and Fb+‘M = 0. ANNAILES SCIENTIFIQUES
DE L’lkOLE
NORMALE
SUPBRIEURE
662
w.
Consider the category M3[+ faithful1 functor L(M)
NIZIOL
(V) with b - a < p - 2. There exists an exact and fully
= ker(F’(M
@ SL(-b}(-b))
‘3
n/r @ Bz(-b)),
where f-b}, (-b) are the M3 and Tate twists respectively, from M31~,~1 (V) to finite Z,-Galois representations. Its essential image is called the category of crystalline representations of weight between a and b. This category is closed under taking tensor products and duals (assuming we stay in the admissible range of the filtration). We will need the following theorem. 2.1. - Let X be a smooth and proper scheme over V = W(k) of pure relative dimension d. For d 5 p - 2, H,“,(X/V, Oxlv/p”) lies in M~L~,,I(V). THEOREM
It was proved by Kato 217, 2.51 (the projective case), by Fontaine-Messing and by Faltings [5, 4.11. Here,
[II,
2.71,
is equipped with the Hodge filtration PH,“,(X/V,
OX,~/~“)
= Im(H”(X,,
$&J
+ H”(X,;
GJ~,)),
and the mappings 4” = 4*/p”
: F”H&(XlV,
Q&P”)
+ H,“,(X/V,
OX/V/P”),
where 4 denotes the crystalline Frobenius. 2.3. Syntomic regulators. Let X be a syntomic, that is, a flat and local complete intersection scheme over W(k). Recall the differential definition [17] of syntomic cohomology of Fontaine-Messing [ll]. Assume first that we have an immersion i : X c) 2 over W(k) such that 2 is a smooth W(k)- SCh eme endowed with a lifting of the Frobenius F : 2 --f 2. Let D, = Dx, (&) be the PD-envelope of X, in 2, and Jo, be the ideal of X, in D,. Consider the following complexes of sheaves on Xzar
where 4’ is the “divided by p”’ Frobenius. The complexes s,(T)x, sh(r)x are, up to canonical quasi-isomorphism, independent of the choice of i and F. In general, immersions as above exist locally, and one defines S,(T)X E D+(Xzar, Z/p”) by gluing the local complexes. Finally, one defines Safe E D+((X,),,,, Z/pm) as the inductive limit of s,(T)x”, , where V’ varies over the integral closures of V in all finite extensions of K in E. Similarly, we define sI(r)x and sk(r)x,. One sets Hi(X, 4e
Sl?RIE - TOME
S,(T)) := H&,(X, 31 - 1998 -
No
j
s,(T)~),
Hi(Xv,
G(T)> := H;,,(+,
sn(~)x~).
663
CRYSTALLINE CONJECTURE VIA K-THEORY
We list the following properties: (1) There is a long exact sequence
(2) There exists a well-behaved product
s&-)x RL S,(T’)X -+ Sn(T+ T/)x;
T + T’ L p - 1.
Since, for X smooth and proper over W(k) of relative dimension less than p, ~~,Fww),
Qx,/w(k)lPn) Z~XI;;IW~),
= Jfc,(wv4,
49
QX/W(k)/Pn)
= ~‘vcr(xI~(~),
@ B,+,,
~X/W(k)lP”)
[ll, 1.5,]
63 B,‘,)
Property (1) yields a natural map Hi(Xp,
&(r))
4 L(@,(X/W(k),
for
0x,w(lc,/p”){-r})
For a scheme X, let K,(X) ble the higher Quillen [ 191. Similarly, f or a noetherian scheme The corresponding groups, with coefficients Z/t and KL(X; Z/Z). Unless otherwise stated, we will K*(X; Z/Z) is an anticommutative ring. For a noetherian regular scheme X, set
p - 2 2 T > dimXK.
K.-theory groups of X as defined by X, let K:(X) be Quillen’s K/-theory. [24], will be denoted by Ki(X; Z/Z) assume that I # 2,3,4,8. In that case,
’ G(X) = i . (yi, (21) . . . yin (xn) I&(x1) = . . . = e(xn) = 0, il + . . . + i, > k, ) F,Kq(X; Z/p”) = (aq&~~) u . ’ . uyin(~,)Ja~~~Ko(X),io+il+...+i,
ifj 2 0, if j > 0,
2 k,),
where E is the augmentation on K,(X). We will use the following theorem of Gros [15]: THEOREM 2.2. - For any syntomic W(k)- SCh eme X and any integer 0 5 i < p - 1, there are functorial Chern classes
cy
: Kj (X) -+ H2+qX,
s&))
compatible with the crystalline Chem classes
Gros’ construction ffollows the method of Gillet [13], which can be extended easily [22] to yield functorial Chern classes ?Syn 2.7 : Kj(X;
Z/p”)
-+ @-j(X,
compatible with the classes c;sjy”. ANNALES
SCIENTIFIQUES
DE L’kOLE
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SUPeRIELJRE
sn(i))
for j 2 2 and j = 0,
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W. NIZIOL
LEMMA 2.1. - We record the following
1. 2. 3. 4.
properties of syntomic Chern classes: cZyn, for j > 0, is a group homomorphism. E%T, for j 2 2 and P # 2, is a group homomorphism. i$‘” are compatible with the reduction maps sn(i) -+ sm(i), n 2 an. Zf u: E Kl(X; Z/pn) and Q’ E K,(X; Z/p”), then c$n(cm’)=
(i - l)! - c E;~(a)c~;“(d): r+s=i (7. - l)!(s - l)!
assuming that 1; q > 2, 1 + q = j, 2i 2 j, i 1 0, p # 2. Moreover, if X is regular, then 5. If a E FJKo(X), j Z 0, and o! E FtKq(X; Z/p”), q 2 2, P # 2, then
E;$&Kd) = - (i(l:)~(kl)j)!c~~n(n)il;:n(a’). Similarly, for cJ?y&, if a’ E F,K,(X), k # 0. 6. The integral Chern class maps cam” restrict to zero on F;+‘Ko(X), i > 1. 7. The Chern class maps 2:;” restrict to zero on FG’lKj(X; Z/p”), j > 2, p # 2. Proof. - Recall the construction (due to Gillet [13]) of the higher characteristic classes E”“. The fastest way to describe it is as follows. One constructs universal classes Ci:” E Hzi(B.GL~/W(k),s,(i)) (de Rh am classes of the universal locally free sheaf on B.GLr,,/W,(k)). They yield compatible universal classes c$” E Hzi(X, GL&‘x), s&)), h ence a natural map of pointed simplicial sheaves on X, C;Yn : B.GL(Ox) -+ K(2i, S,(i),y), where K is the Dold-Puppe functor of r$&(i)~[2i] and s”lz(i)x is an injective resolution of sn(i)x. The characteristic classes i$‘” are now defined [13, 2.221 as the composition KJX;
Z/p”)
---f H -j(X,
Z x B.GL(&)+;
H-Yn)H-qX,
Ic(2i, 3&)x);
Z/p”) Z/p”)
-+ H-j(X,B.GL(Q)+; J-3 IF-qx,
Z/p”)
.s&)),
where B.GL(Ox) + is the (pointed) simplicial sheaf on X associated to the + -construction [21, 4.21. Here, for a (pointed) simplicial sheaf E. on X, H-j (X, 8.; Z/p”) = rj (qx, -2.); Z/P”) is the generalized sheaf cohomology of E. [13, 1.71: if we let ?$ denote the constant sheaf of j-dimensional mod p” Moore spaces, then H-j(X,E; Z/p”) = [T-J;; &.I, w h ere, for two pointed simplicial sheaves F,, F.’ on X, [F,, .F!] denotes the morphisms from F. to .F’ in the homotopy category. The map f is defined as the composition P(X,
Ic(2i, sxn(i)x); Z/p”) = 7q(K(S -S&)(X)>; Z/P">) 2 H,(K(2i, s”&)(X)); -+ Hj(CK(2i, S,(i)(X))) c Hj(NK(2i, S”&)(X))) = H&(i)(X)[2i]) = IF-qx, &(i))
Z/p”))
Here, hj is the Hurewicz morphism, and, for a simplicial abelian group A, CA and NA denote respectively the associated chain complex and the normalized chain complex. 4e
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K-THEORY
Since, for p # 2, the Hurewicz morphism is compatible with addition and products [22, 11.2.21, so is the map f. The classes c:T are constructed in an entirely analogous manner. Since the map A,, : @ ---f ?$ A 7’5 is nullhomotopic for odd p, properties 1 and 2 follow as in [13, 5.261 from the Whitney sum formula for the syntomic classes of representations of sheaves of groups, which in turn follows from the fact that the de Rham cohomology of B.GL(n, m)/VV(k) is nontrivial only in even degrees (hence the even-degree syntomic cohomology injects into the de Rham cohomology) and the Whitney sum formula is true in the de Rham cohomology. Property 3 is easy to check. For the remaining properties, consider the following diagram PO B.GL(&)+ B.GL(&)+ A B.GL(CJx)+
Here, CSY” is the total Chern class, cl0 is the Loday multiplication, and * is the Grothendieck multiplication [2, 0.31. We claim that the above diagram commutes. This follows, as in [13, 2.321, from the tensor product formula for the syntomic classes of representations of sheaves of groups, which in turn follows, just like the Whitney sum formula above, from the fact that the de Rham cohomology of B. (GL,/W(k) x GL,/W(k)) is nontrivial only in even degrees. We have [20, 2.3.41 (C~;)*(~Yi)=C~l(alj...:~z,Yl;...;Yz), i>l
i>l
where ~(~~,...,~z,Y~,...,Yz)
120
=
(C,+,=z
G-~G-Y~
+
&(~)T,(Y))
and
G-.~
=
-(I
-
l)!/((r - l)!(s - l)!). H ere, 2, and 97, are polynomials of weight T and s respectively, and iE T + s = 1 then either 2, or T, is decomposable, i.e., can be represented as a sum of products of monomials of lower weight. To prove properties 4 and 5, it suffices to show that the terms &(z)T,(y) disappear when evaluated on the corresponding K-theory classes. For any q E Kj(X; Z/p”), j > 2, and any a, b 2 1, the following diagram commutes
-
B.GL(Ox)+
B.GL(Ox)+ A
ab
a6 1CW”ACW”
CSY” ,CW” 1
fq2(u
+ b), &(a + b)x)
Thus, since A,$ get Property 4.
is nullhomotopic,
ANNALES
DE L’lkOLE
SCIENTIFIQUES
NORMALE
=
A B.GL(Ox)+
q2a,
&(4x)
A lq2b, S,(b)x).
the decomposable terms 2, or T, disappear and we
SUP6RIEURE
466
W. NIZIOL
Assume now that we know that the Chern classes in properties 6 and 7 vanish on the monomials rk(z), k > i f 1. Notice that it suffices to prove properties 5, 6, and 7 for the products of monomials ye (z) : the general case will follow from the Whitney sum formula and property 2. We will thus assume now that all the K-theory classes are products of monomials as above. We will argue for property 5, the case of o’ E F$KO(X), k: f 0, and property 6 simultaneously, by induction on the number of monomials Ye in the appearing K-theory classes. If we use S/g (the simplicial version of the O-sphere) instead of F’i above, since (by induction from property 6) CzY”(a) = 0 for a < j (and similarly for a’), we get the vanishing of the terms Z,T, we wanted. Property 6 follows now by induction from the multiplication formula in property 5. We will argue for property 5, the case of cx’ E F$K,(X; Z/f), by the same type of induction. By induction from properties 6 and 7, all the terms Z,T, disappear. Property 7 follows by induction from the multiplication formula in property 5 and from property 6. It remains to show that 2:;” = 0 on elements of the form Yk (x), k > i + 1, II: E Kj(X; Z/p”) (and an analogous fact for the integral K-theory classes). We will reduce to a similar statement for the equivariant Chern classes of group representations. ecall how the y-operation Tk acts on Kj(X; Z/p”) [23, 41. Write K(GLN) for the Grothendieck group of representations of the group scheme GLN over Z. There is a morphism of abelian groups ,rAr : K(GLN)
-+ [Z x B.GLN(Ox)+,
Z x B.GL(Ox)+]
induced by a transformation, which to every representation p : GLN + GLM of the group scheme GLN associates a morphism of (pointed) simplicial sheaves B. GLN(13x) 4 B. G,?LM(0,). Passing first to the morphism induced on generalized cohomology groups, then to the limit over N, we get a morphism of abelian groups T : K(GL)
-+ Hom(Kj(X;
-+ Kj(X;
Z/p”)
Z/f)).
Now, the operation O/kis defined as the image under T of the element (Tk(IdN - N))N E K(GL), where Id,v, 1V are the classes of the natural, respectively trivial, representation of GLN. Prom the above description it is clear that it suffices to show that the composition B.GLnr(C?s)
3
I?.GL(Ox)+
2
Ic(2i, &&)x)
is homotopically trivial. Set, after Gillet [13, 2.271, N2i(X,
z x B.GL&Jx)+
, s&))
: = [Z x B.GLN(Ox)+,
ii*(X,
Z x B.GLN(Ox)+
, sn(*))
: = H”(X, n
Ic(2i, S”&y)];
Z x B.GLN(Ox)+, Hzi(X,
Z) x (1)x
Z x B.GLN(Ox)+,
sn(i)).
i>l
H*(X,Z
x B.GLN(O~)+,S,(*))
is a X-rmg [ 13, 2.271. Consider now the augmented
total Chern class map E : K(GLN) 4e
SGRIE - TOME
--i ii*(X,
31 - 1998 -
No
Z x B.GLN(&)+, 5
sn(x));
P ++ @4P)>
C.(P))
CRYSTALLINE
CONJECTURE
VIA
667
K-THEORY
We claim that this map is a X-ring homomorphism. Indeed, it is clear that the map is well defined. Compatibility with addition and multiplication follows from the Whitney sum formula and the tensor product formula, which, as we have mentioned above, hold for the syntomic Chern classes. For the compatibihty with X-operations, in suffice_s to pass to the correspoding B.GL,M/W(IE) and check that c(/l\“Z) = A” C(E) E H*(B.GLM/W(~), sn(*)), where E is the universal locally free sheaf on B. GLM/W(~). As before, it suffices to check this formula in the de Rham cohomology of B. GLM/W(k). This can be done easily using the slplitting principle. Returning to the main line of the argument, we have to show that the class of cI~~(~yk(Id~ - N)) E H2i(X,GL~(Ox),s,(i)) is zero. This now is standard. Set XN = IdN -N. The above yields that the Chem polynomial cz:, is a X-ring homomorphism. Hence we have ci;ih(~N))
=
%(c$&V))
=
Yk(l+
c;;;((zN)t
= 1 + (-1)“~‘(k
+
C;y;(xN)t2
- l)!c;T;(xN)t’
+
. . .)
[2, 0.1.21
+ ...
Thus c~~~((r~(~~)) = 0 for 0 < i < Ic, as wanted. Similarly, by recalling [2, 0.1.21 again, we get that the integral Chern class maps cig”, i 2 1, are trivial on ~yk(5), k 2 i .-t 1, for any 2 E KO(.), E(X) = 0. Remark 1. - For any K-scheme
Y, one can paraphrase the a.bove proof to show that
the &ale Chern class’ maps
c$ : Kj(X) + H-(Y,
Z/p”(i)),
E$ : K,(X;
Z/p”)
+ Hy(Y;
Z/p”(i))
have analogous properties to those of the syntomic Chern classes. Similarly, working with the complex S;(T) instead of the complex S,(T), we get coholmology groups Hi (X, .sL(r)) and Hi (Xv, si (r)), which are targets of higher Chem classes [15, 2.31. All of the above listed properties of s,-Chem classes carry over to the s; -theory. For a V-scheme X, let Xa denote the special fiber of X. Recall that, for X smooth and proper over V, Kato-Messing [ 181 have constructed the following isomorphisms
H:,(Xo/W(k),Oxa,w(lc)) @Q @B,+, zr
Q @ projlimHk(Xi;i/W(k);
H&dWK)
k Bj--
E w$WQ
F’(H;,(&c/K)
@ pr0.in limH:,(Xv/W(k),
@B,+,)2
proj lim,(Q
We will need to know that ANNALES
[18, 1.2],
OX,~W(~)/P~)
SCIENTIFIQUES
DE L’!kOLE
NORMALE
SUPtjRIEURE
Ox~,~~Wn!k)lJ~N]) @ proilim
H:,(Xv/W(k),
[18, 1.41, $]/JiN])).
668
w. NIZIOL
LEMMA
2.2. - 1. The following
@ proj limHi(Xv; n
s;(r))
two compositions of maps are equal
+ proj lim,v(Q @ proj lim,H&(X~/W(k),
J~l/J~Nl))
where 6 is induced by the Berthelot-Ogus isomorphism (3, 2.21 Hz,(Xo/W(k)) @W(k) K E H&(&c/K). 2. For any smooth and projective scheme X over V, the Kato-Messing isomorphism her
:
Q@proj limHk&/W(k), 71
OXF/W(k)/pn)
=
Ox,,W(k))@Q@B~
H~,(&/W(lc),
is compatible on the image of the crystalline Chern classes 6:; : Ko(X)
-+ Q @ proj lim H,2,i(X/W(k), 72
ox,W@)/p”)
with the natural map
Q@
PM
lim,HZ,(XIW(~),
oX/W(k)/$)
Q@
-+
H:,(xO/w(@;
oXo,w(k)).
- Proof.- Letv = v @W(k) v, x = x @M(p) V, EC+,=
proj lim,Hzr(Vn,/V,). By the syntomic base change theorem [l, 2.3.51 H,*(V,/V,) N H&(7,/W,(k)) @w(k) V. Hence H&(pn/V,) = 0, for i > 0, and gz = J32 @W(k) V. In particular, 52 is flat over V. Using that and unwinding the definition of the Kato-Messing isomorphism h,,, we easily check that the following compositions of maps are equal
Q@
Q8
problimH~,(X~,n/W,(kj)
proj
% +
@:,(Xo/WV)
2
H:,(Xo,v/V)
limH~,(X~,TL/Wn(k))
H:,(x~/w(k)) @ w(k)
Q@
@v -+
Q@
proj
Q@
B:
@w(k)
Q@
@w(k)
V
BL
g,+,; lim
H$(Xv,n/W,(k))
n
f-
Q @ Hzr(X~/V)
-f+ H:,(Xo;v,‘V>
@V proj lim H,o,(%/V,)
72 @v Q @ @
Here, Xa,v = X0 @w(k) V, the map AZ is the base change morphism, and the map Xi is the Ktinneth morphism. This yields Property (1) of the lemma. Concerning Property (21, since the morphism
Xo: H:,(XoIWW @Q @B,t. -+ H:,(Xo/W(W @w(rc) Q @B; @w(k)V 4e
SERIE - TOME
31 - 1998 -
No
5
CRYSTALLINE
CONJECTURE
VIA
669
K-THEORY
is injcctive, it suffices to show that the composition
is the natural map H&(X1/W(k)) that tlhe composition
-+ H&(X,/V)
on the level of Chern classes, and
is compatible with the natural map H&(X/V) ---f Hz, (Xr /V) . The latter assertion is easy to show, and the former one follows from [3, 3.71.
3. K-theory
lemma
Let V be a complete discrete valuation ring with a field of fractions K. Let v denote the integral closure of V in F. LEMMA 3.1. - Let X be a smooth V-scheme. For any integer n, the natural morphism j* : Ki(X @V V; Z/n) + Ki(X @J,JK; Z/n) is an isomorphism. Praof. - We may assume the residue field of V to be separably closed. Let L/E/K be finite field extensions with rings of integers V, and If’, respectively. Let e be the ramification index of L/E. The map f : XV, -+ XV, being flat, the localization theorem in Ill-theory yields the following commutative diagram of maps of exact sequences
KI+,(XG Z/n)
-
Ki(X, @v, T/i,;Z/n)
-
q(X”,;
Z/n)
-
K;(xL; Z/n)
fg=,i* I T T Z/n)- q(XJ/,; Z/n) __fIC,'(XlJ; Z/n) q+,(xE; Z/n) - IqXK; flu,
* fx,
fXVET
k
The scheme Xz@vE I/L is non-reduced, with underlying reduced scheme XT, and we write i for the inclusion Xx c-+ Xx @vE V& We claim that fg- = ei,. Indeed, let M(Z) denote the exact category of coherent sheaves on a scheme 2” Then f-g- = f,” for the functor fK : M(X,) -+ M(Xzgv, VL), fK(F) = ;F@, VL. Since x@T:EkVLN $z]/se, filtering fK, we compute that f,” = CyzI J$, where every functor fz : M(Xz) -+ M(Xx@v, V’) is naturally equivalent to i,. From that, we easily see that there is a natural isomorphism inj lim K,((Xv,; LIK
Z/n) -% inj lim,,,Kl(XL;
z/n),
where the limit is over finite field extensions of K. Since the original scheme X was smoo’th, all the schemes XV, are regular, and we can pass to K-theory in the above isomorphism getting the statement of our lemma. ANNALES
SCIENTIFIQUES
DE L’lkOLE
NORMALE
SUP6RIEURE
W. NIZIOt
670
4. Integral
theory
Let the notation be that of section 2. Let X be a smooth quasi-projective V-scheme of relative dimension less than p - 2, V = W(k). We would like now to construct functorial Galois equivariant morphisms &zb
: H”(XE,
z/p”(b))
+
L(H,“,(x/v,
sX/P”){-b)).
We will be able to do it (4) only under certain additional restrictions on the integers a, b and d. Our construction will be based on the following diagram F@?;+lKj(XT;
Z/p”)
+
Z/p”)
F$‘F;+lKj(XK; 3
H”“-qxv;
s&))
H2i-qxz,
Z/p”(i)).
Here 1 5 i < p- 1, j 2 3, p” 2 5, p # 2. The Chern class map Cri” : F;Kj(Xv; Z/p”) + IS Hz”-j(X+&)) . d efi ne d as a limit over finite extensions V’/V of the syntomic Chern class maps F$ Kj (Xv) ; Z/p”) -+ Hzi-j(Xv,, s,(i)). Due to Lemma 2.1, the Chern class maps E$ and E:;” factor through FG’l yielding the maps in the above diagram. We will need to know how Bott elements in K-theory behave under higher Chern class maps. Recall that to every sequence of nontrivial p-roots of unity & E qPL
o-+K~(Q~; z/p”)3 K,(Qp)c WQp)-+0. We will denote by ,& the unique element in K2 (&, ; Z/p”) mapping to ,& E K2 (Q,; Z/pn) (such an element exists and is unique by Lemma 3.1). We have a commutative diagram
d
K2(z,;
Z/P”)
-
K1(-Z,)
“,
where all the vertical maps are injections. Hence a(,&) LEMMA
$
= & as well.
4.1. - We have C”ly;(i,)
= t E H”(-z,, $:203n>
sn(l)),
L$$@n)
= CmE H’(Q,;
= t E Ho&,, Z/P”(~)),
where t E Bz( Z,) is the element associated to & *s (c$, section 2.1). Proof. - By SOL% [22, IV.1.31,
$2 : K2(Qp; Z/P") --f H’(Q,, 4e SBRIE - TOME 31 -
1998 - No 5
Z/P"(~))
s;(l)),
CRYSTALLINE
CONJECTURE
VIA
K-THEORY
671
is the Bockstein map followed by 1:he determinant map:
For the syntomic class, notice that we have an injection H”(zp, sn( 1)) ci ~,q.((-Z;,/P”wn(~), AL>. S’mce the syntomic classes and the crystalline classes are compatible, it suffices to show that Ed, = t E H~~((?&/p”)/VV~(k), Jn) N FlB,S,/p”. Consider the Kummer exact sequence n
0 + Z/p”(l)
+ G, % 6,
+ 0
on the Zariski site of Spec@,). Define classes of the natural representations ci(Idk) as the image via the boundary malp f%w(z,, GL@p), Z/V(l))
E
of the class of the GLk&,)- invertible sheaf associated to det (Idk). Paraphrasing the above mentioned argument of Soul6 one can show that, if we induce a map Cl,2
: K2(z,;
Z/P”>
-+
G,,(z,,
Z/P”(l))
via m’od p” Hurewicz morphism and via the Ktinneth homomorphism
from the classes cl(Idl,), then ~~,a(,&) = &. Consider now the following diagram
Kdzp; VP”)
where u : (Spec(&,/p”)/W,(k)),, -+ Spec(Z,)z,, is the projection from the crystalline to the Zariski topos and the map cv is defined by sending & to t on the open Spec(%,) and & to 0 on the open 62p. It suffices now to prove that this diagram commutes. Since both maps c1,2 and Eit2 are defined via the same procedure from characteristic classes of repres,entations (to see this for the crystalline classes, adapt the argument of Shekhtman [20, 2..2] to K-theory with coefficients), it suffices to prove that the map
maps cl(Idk)
to cy(Idk).
But the last classes come via Bockstein maps
G, 5 Z/p”(l)[-111, ANNALES
SCIENTIFIQUES
DE L’kOLE
NORMALE
G, SJPliRIELJRE
+ Gm,Z,pn l”~r~*&[-l]
672
w. NIZIOt
from the class of the GLI,(‘Z,)-’ mvertible sheaf associated to det(Idk). Here, d, 8”’ are the boundary maps coming from the Kummer sequence and from the exact sequence 0
+
(1 +u*J?Jzp)
-
(b*QJ(z,H*
il 0
-4
-+
i
0
il
II
+ (B&/p”)*
1 + FlB,S,/p”
(Z/pn)*
-+
Eh”)*
+
0,
respectively. Hence, it suffices to show that the following diagram commutes tin the derived category) d Gn %wN-11
It is clear on the open Spec(?&). For the commutativity the following commutative diagram with exact rows 0
+
-+
1+ FlBQp” T
0
i
on the open Spec(Z,),
(B&/p”)*
-+
+
--f
0
+
0
T
,7T
proj lim pPn(Zi)
(Z/pn)*
consider
proj lim Zi
--tp’
55;
where the maps in the projective limit are the p-th power maps, and the map y sends a sequence ( ui) to [(a;) 90, . . .] . This diagram yields the following commutative diagram
(z/p)*
5
(1+ F’B,s,/p”)[-l]
II zg
+
a
proj lim/+
F1%IP”I-K
(57;) [-l]
II
1 /App”(z;)[-l]
FlB,+,/p”[-11
II
T
21T
55; + a
J5
“,
J%+JP”[-11,
which completes our argument. For the s’-class, since the syntomic classes and the crystalline classes are compatible, it suffices to show that Cy:z(Dn) = t E H,q((zp/p”)/Wn(~),Jn) rv FlBL/p”. This was done above. In what follows, we will fix a generator < = (&) of Z,(l) and denote by ,&, jn, and t the elements corresponding to C. For a scheme Y over Z,, let &, E K2(Y; Z/p”) denote the pullback of the Bott element jn E Kz(z,; Z/p”). We will denote by Pn the image of pn in K~(YQ ; Z/p”). For a positive integer d, let N(d) denote the constant defined by Thomason inP[24]: N(d) = (2/3)d(d + l)(d + 2). 4" SI?RIE - TOME 31 - 1998 - No 5
673
CRYSTALLINE CONJECTURE VIA K-THEORY
PROPOSITION 4.1. - Let Y be a smooth quasi-projective scheme over -ii’. Let p # 2 and n be such that p” 2 5. Let i be any positive integer. There exists a constant T = T(d, i,j) depending only on the dimension d of Y and i, j such that, for j 2 2N(d), 1. the cokernel of~$ : FG/Fi+lKj(Y; Z/p”) + i!?j(Y, Z/p”(i)) is annihilated by T; 2. iv [x] is in the kernel of C$, then T[x] = [z] for z E FiKj(Y; Z/p”) annihilated by /3:(d). Moreover, for p > max{i + 1; (d t j + 3)/2}, we can take T = 1.
Proof. - The proposition follows from Thomason’s comparison theorem [24] between algebraic K-theory and Ctale K-theory, and from the degeneration of the Dwyer-Friedlander spectral sequence modulo torsion [21]. We get a uniform constant T as in the theorem through a careful bookkeeping of the appearing torsion - most of the work being done by Soul6 in [21]. Soul6 [21,4.1.4] has shown that tlhe algebraic Chern class c$ is equal to the composition of the topological Chern class cij : KF(Y; where Kp(Y; natural map
Z/p”)
Z/p”)
-+ IP-qy,
is the &ale K-theory
pj : K,(Y;
Z/p”)
from algebraic to &ale K-theory. Recall that we have a Dwyer-Friedlander E2S,4=
i > I, j > 2,
Z/p”(i)),
of Dwyer and Friedlander [4], [12], with the
+ KfyY;
Z/p”)
spectral sequence
H”(Y, Z/p”(i)) { 0
if 0 5 s 5 4 = 2i, otherwise
converging to Kp-,(Y; Z/p”), 4 - s > 3. Let F”KF(Y; Z/p”) denote the filtration on Kp(Y; Z/p”) defined by this spectral sequence. Consider also the following y-filtration F; = (y”(x)lk 2 i). S’mce the X-ring multiplication on KY (Y; Z/p”) is trivial, this version of y-filtration satisfies many of the formal properties of the filtration F;. In particular, we can prove, as in SoulC [21, 3.41, that M(d; i,j)F2i-jKF(Y;
(3)
Z/pn)
c F;KF(Y;
Z/p”)
c F2i-jKp(Y;
where the constant M(d, i, j) is to be defined below. NOW, we know [21, 4.21 that cij restricts to zero on Fzi-j+lKp(Y; inducles a map cij : F;/j?FlKF(y;
Zip”)
L!+ J$--j /F2i-j+2Ky(Y;
Z/p”),
Z/pn).
Z/pn) 3 l?-j(Y,
Hence, it
Z/p”(i)).
We will now study the maps f, y. Let’s first recall some constants from [21, 3.41. Let 1 be a positive integer, and let wl be the greatest common divisor of the set of integers kN (,@ - I), as k runs over the positive integers and N is large enough with respect to 1. Let M(k) be the product of the ~1’s for 21 < k. We will also need the integer M(k
ANNALES
m,
n)
=
r12m<21<2m--n+k+l
SCIENTIFIQUES
DE L’lkOLE
M(24.
NORMALE
SUPkRIEURE
474
W. NIZIOL
By the inclusions (31, the map f has kernel and cokernel respectively AI( d, i, j). Concerning the map g, notice that, by Soul6 [21, N2i-j(Y, Z/p”(i)) = Eii-“2i lies in the kernel K2i-j~2i T > 2, in the Dwyer-Friedlander spectral sequence. Hence, g : F~~-~/F2~-.i+2K~(yj
Zip”)
+ Kzi-izi
annihilated by M(d, i + 1, J’), 4.21, the image of cij in of all higher differentials d,, we have a factorization
L+ Hz”-j(y,
Zip”(i)).
Since [21, 3.3.21 M(d)& = 0 for any T > 2, the cokernel of the inclusion K2i--ja2i q H2i-j(Y, Z/p”(i)) is annihilated by M(d). Consider now the composition E2i-j,2i cc
,$-j/j@-j+2Kft(y; h
J$i+,F%j+2Kj”t(y;
Zip”)
i Zip”)
pi2i-j,2i =
$72~‘j,2i,
where $;j is the natural projection. This composition is proved in [21, 4.21 to be equal to multiplication by (-l)i-l(i - l)!. H ence the kernel of g is annihilated by (i - 1) !. Also, since &f(d)&. = 0 for any T > 2, the kernel and cokernel of & are annihilated by n/i(d). Hence the cokernel of g is annihilated by M( d)2 (i - 1) !. The above yields that the kernel of Cij is annihilated by Ill(d, i + 1; j)(i - I)! and its cokernel by &I(d;i,j)IU(d)‘(i - l)!. Consider now the map
We know from,Thomason [24], [25, 3.21, that ,5j is-surjective for j > 2N(d). For its kernel, let 5 E FGK,(Y; Z/p”) be such that pj(z) E F$+lKp(Y; Z/p”). If j > 2N(d), there exists a y E F;+lKj(Y;Z,p’“) such that pj(z) = pj(y), or pj(z - y) = 0. Hence, if we set z = IC - y, then y = z - z E F;+‘Kj(Y; Z/p”) and z is annihilated by @r(d) (again by Thomason). Combining all of the above, we get that, for j 2 2N(d), the cokernel of E$ is annihilated by M(d, i,j)M(d)‘(i - l)!, and its kernel, restricted to F;Kj(Y; Z/p”)/@G+lKj(Y; Z/p”), is, modulo M(d, i + l,j)(i - l)!, annihilated by ,&fcd). Since M(2i)F;Kj( Y; Z/p”) c FGKj(Y; Z/p”) + F:+lKj(Y; Z/p”) (argue as in [21, 3.4]), we can take T = T(d, i,j) = (i - l)!n/i(2i)AJ(d, i,j)AJ(d, i + 1:j)M(d)2. Since an odd prime p divides &I(d, i, j) if and only if p < (J’ + d + 3)/2, and divides 0 M(Z) if and only if p < (l/2) + 1, we get the last statement of the proposition. Remark 2. - The results of Soul6 needed in the above proof are stated in [21] for the p-adic topological K-theory. Working with Friedlander’s definition of &ale K-theory mod pn 1121, i.e., using systematically P”(p”) A Xkt, where P”(p”) is the simplicial Moore space obtained from the mapping cone of multiplication by p” on the circle and where, for a scheme X, X6, is its &ale topological type, instead of Xbt, one sees that Soule’s results hold as well for the p-torsion K-theory (with the usual restriction on the degree of the K-groups) and his proofs carry over to that case almost verbatim. Let now p, a, b, d satisfy the following condition
(4 4e
b 2 d, p - 2 2 max{b + N(d), (1/2)(d S&RIE - TOME
31 - 1998 -
No
5
+ 2b - a - l)},
2b - a 2 2N(d).
CRYSTALLINE
CONJECTURE
VIA
675
K-THEORY
Define the morphisms %b
: Ha(Xz,
z/p”(b))
-+
L(H,“,(x/v,
oX,v/~“){-b~)
as the composition oab := ~,C~~bb--a(j*)-l(C~~~b-a)-l, where &, is the natural map H”(XF, s,(b)) * L(ffFr(X/V, CJ&v/p’“){--b}). Here (2””b,a&a)-l (x) is defined by taking any element in the preimage of x. We do not know whether this is well-defined, but we do know that its composition with q!~~~~~~-~(j*)-’ is. Indeed, by the above proposition, an ambiguity in the definition of (el;b-a)-l may only come from a class of z E E~=&-a(X~; Z/p") such that pzCd)z = 0. So bt’“‘(j*)-l(z) = 0. Hence
’ = ~~[~~~(d),zb+2N(d)-~~~~~(d)(j*)-1(r))l = (-l)N(d)(N(d)
+ b - l)!/(b
- l)!~n[~~~(~~(d))~~~~b-~((j*)-l(~))]
= (-l)N(d)(N(d)
+ b - l)!/(b
- l)!~“‘“‘?i,l~~~~b-~((~,~)-l(~)).
To conclude that $~~Ci~t~~,((j*)-~ I(z)) = 0, as wanted, it suffices now to show that, for b + 1 < p - 2, mult$ication by t
is an isomorphism.
But the multiplication
map
induces a map L(1Z&(X/V,
ox/v/P”){-b})
8 L(V/P”{-1))
+ L(JJ,“,(XIV,
which, since H$(X/V, 0 x,v/p”){--b - l} E MFL-b-i,a](V) isomorphism. Since multiplication by t yields an isomorphism [9, 5.3.61, hence an isomorphism L(FxX/V
-4, L(q$qV
Qx/vlP”){-w
Ox/v/P”){-b})
CC/v/P”){-b
- I>)>
(Theorem 2.1), is an Z/pn 2 L(V/p”{ - 1))
@ L(V/p”{-l}),
we are done. Assume now that X is projective of pure dimension cl. We will show that in that case the maps o&b are isomorphisms. First we prove 4.2. - 1. The maps o&b commute with products. 2. The maps c&b are compatible with some cycle class maps, i.e., if Z is an irreducible closed subscheme of codimension j in X, smooth over V, then
LEMMA
aQ,b(d”t(zK)[b-‘) = d=‘(Z~,)t~-~~. Here clCr(Z,) E H,“,j (X,/V) ANNAL.ES SCIENTIFIQUES
DE L’kOLE
NORMALE
is the one dejined by Gros in [ 161. SUPkRIEURE
676
W. NIZIOL
3. For irreducible X5
the following H2”(X,,
diagram commutes
Z/p”(b))
q+
ZlPV
piI
4d,b I
- 4
Proof. - Property 1 follows from the multiplication formula in Lemma 2.1.5. For property 2, let [0,] and [(3zK] denote the class of 0~ and OZK in Ka(X) and K~(XK) respectively. From the hypotheses, both 2~ and .Zk are integral and of codimension j in their respective fibers. We will need to do some preliminary computations. Let s(d) = fli?$: AY!.First, we claim that s(d)[0,] E F$Ka(X) (it follows then that s(d)[0,,] E F$Iyo(XK)). By the Integral Riemann-Roth [23, 4.61, for any z E F$&(Z) T;~($+(.z)) = i,((-l)‘+“-l(~ + 1 - l)!z + y), where i, : Ka(Z) + Ka(X) and ‘+lKo(Z). Since F ‘+‘K O(2) = 0, this gives our claim. NeXT, we claim that, for 0 < i < j, the Chern classes c$a([OzK]) and c~~([U3,]) are zero. For the Ctale Chern class, since for an open U in XK such that the codimension of XK \ U is strictly greater than i, the restriction H~~(XK, Z/p”) -+ Hij( U, Z/p”) is an injection, we may assume that ZK is the zero-scheme of a regular section s of a locally free sheaf E of rank j on XK. The Koszul complex gives a resolution
Hence [0,,] = C(--l)‘[Ai(c)] in Ka(Xh’). 0 ur claim now follows from [2, 0.1.21. For the syntomic Chern class, notice that $~~cfridl([13,]) = c~~~([C?~,]). Recall that the crystalline Chern classes factor through the logarithmic de Rham-Witt classes ~~~lfo([~Z,I) E @(XI,, wLf&k,log ). Hence it suffices to show that C;:a(IO,,]) = 0. This can be shown exactly as in the &ale case (for the injection of the restriction map @(Xk: wnqyk,log ) --+ Hi(U, Wnfl&,lOg) refer to the purity result in the logarithmic de Rham-Witt cohomology [ 16, 11.4.1. I]). Finally, from the Whitney sum formula we have
= 4~)~ncg~;;,^([~z]). ~;~owwZKl)= 4~)c~~il([~Z,l), ?k;~~b”(~(~)~~z1) We are now ready to prove property 2. Since c$@,) = & (& corresponding to &) we get from the multiplication formula in and c$([OZK]) = (-l)j-“(j - l)!clbt(ZK), Lemma 2.1.5, we get that --Bt C6,2(b-j)(S(d)(C?ZK]P~-j)
= (-l)bqJ
- l)!s(d)cl”t(&)&j.
On the other hand, ?i/~i$~~(&) = t and &cg~~([Sz]) IV.3.1.21. Hence 1cl,li~~~~6-I)(s(d)[~z;~~-i)
= (-l)b-l(b
- TOME 31 -
1998 - No 5
[16,
- l)!s(cqclc’(z,)tb-~.
By the definition of the map d!aj,b, since our p > max{d+j 4e &RIE
= cjcfO([OZk:]) = clc’(ZI,)
- 1, b- 11, this gives property 2.
CRYSTALLINE
CONJECTURE
VIA
677
K-THEORY
Concerning property 3, we can assume that the residue field of V is algebraically closed. Let f’ be a rational point of X over V. Recall that tret and trC’ are characterized by the mapping ~1”~(PK ) C-“, respectively cl”‘(E’k), to 1. It suffices thus to show that Q;d,b(cl”t(P&-d)
= clcr(P#-d,
which is a special case of property 2.
0
4.1. - For any projective, smooth V-scheme X of pure relative dimension d, the Calois equivariant morphism THEOREM
is an isomorphism, if the numbers p, b, d sati&
b 1 2d + N(2d), p - 2 > 2b + N(2d).
Proof. - We may assume that k is algebraically closed and that X is irreducible. The line of the argument is standard [ll], [6, 2.41. Both the target and the domain satisfy Poincare duality:
Ha&,
Z/p”(b)) 63H”“-‘“(X,, 2 H2d(Xz,
wf,“,(wK
Z/p”(2b))
Qc/vlP”)W))
-5 L(H;f(x/v,
Z/p”(b)) tz Z/pn(2b - d),
(23WL, ““-“(x/v,
0 x,v/p”){-2b})
~x/v/P”){-b})
5 b(V{-2b
+ d}) z Z/p”(2b
- d)
By L,emma 4.2, c&b has a left inverse 0,-d-. To show that GE;: is a right inverse as well, it would suffice to show that it commutes with products. Since it commutes with Ktinneth products (because CY,~is functorial and commutes with cup products), it suffices to show that it commutes with A*, A : X c+ X x X being the diagonal embedding, or equivalently, that c&b commutes with A,. That, in turn, would follow, if we would show that o,b respects class of the diagonal and that A* is surjective. But this holds, since piA = Id, where p1 : X x X + X is the projection onto the first factor, and we have Lemma 4.2. 0 Remark 3. - The lower bound on the prime p appearing here is of order d3, while Faltings in [5] gets an order of d. While it is perhaps possible to decrease our lower bound slightly by a more careful argument, the main obstruction comes from the constant N (d:) of Thomason. It seems reasonnable to believe that the optimal N(d) should be of order d as well.
5. Rational
theory
Throughout this section, let the prime p be odd. Let X be a smooth quasi-projective V-scheme, where now the ring V is possibly ramified over W(k). For large b and p # 2, we will construct Galois equivariant functorial morphisms
ANNALES
SCIENTIFIQUES
DE L’lkOLE
NORMALE
SUPliRlEURE
678
W. NIZIOL
Most of the work was already done in section 4. For j > 3, p” > 5, p # 2, since Lemma 2.1 holds in this context as well, we get the following diagram F;/F;+l
For b > d + N(d),
Kj (X,; Z/p”)
+
3
F;/F;“Kj(Xi7;
Z/pn)
define the morphisms
as the composition
where $J~ is the natural projection
and we set I(b; d) = (-i)“(d)(N(d) + b - l)!/(b - I)!. Here (C&-a)-l(~) is defined by taking any element in the preimage of T(d; b, 2b - a)z (by Proposition 4.1, T(d, b, 2b - a)~ lies in the image of T$&-~ >. Again, by Proposition 4.1, any ambiguity in that definition comes from a class of y such that T(d, b, 2b - a)[~] = [z] and z E Kzbpa(XK; Z/p”) is annihilated by /3zCd). Hence
= ~n[~~~(d):2b+2N(d)--a(~~(d)(j*)-1[Zl)1
= o
and the map atb is well defined. Let now p # 2 and b 2 d + N(d). Define the morphism
p(b))-+ fCXfolW(O Qxo/w(q) @w(k)Rx{-b} as the composition of Q @ proj lim,aEb with the Kato-Messing isomorphism @pi = H,n,(Xokk),
iim~~(X~/W(IC>,O~,/w(k)/p~) Qx,,w(~))
@w(lc) K'i
@Q
and the division by Z(b, d)T(d, b, 2b - a)‘t”cd). THEOREM 5.1. - Let p # 2 and let X be any projective smooth V-scheme of pure relative dimension d. Then, assuming b 2 2d + N(2d), the morphism
is an isomorphism. 4e S~RRIE- TOME 31 - 1998 - NO 5
CRYSTALLINE
CONJECTURE
VIA
679
K-THEORY
Moreover, the map Qi,b preserves the Frobenius and the action of Gal(K/K), extension to BdR, induces an isomorphism of @rations. Proof. -- We have the following
analogue of Lemma 4.2.
LEMMA 5.1. - 1. Let z E H”(Xx:, i(b, d)Z(e, d)K(b,
and, after
Z/p”(b))
and y E H”(XF,
e)t N(d)T(d, b, 2b - c~)~T(d,
= l(b + e; d)K(b,
e)T(d,
Z/p”(e)).
e; 2e - c)~cc~+~,~+~(z
Then U y)
b + e; 2b + 2e - u - c)2azb(x)
u aze(y),
where K(b,e) = (b + e - l)!/(b -- l)!(e - l)!, assuming that all the indices are in the valid range. 2. The maps C&b are compatible with some cycle class maps, i.e., if Z is an irreducible closed subscheme of codimension j in X, smooth over V, and b 2 j, then n2&1”“(Z#-9
= clc’(Z,,)tb+.
Here cl”‘(Zk) E H,“,j(Xo/V) is the one defined by Gros in [16]. 3. For b 2 d and irreducible X;k, the following diagram commutes
Q&W
H2d(X~T,
-
tP
Q&b - 4 p--d
~Zd.26 1 H,2,d(&/WJc),
Qx,,w(&-2b)
1 @ Bcr
trc’
Bcr{d - 2%
Proof. - For properties (1) and (2) we evoke Lemma 4.2 and argue as in the proof of Lemma 2.2.2. For property (3), take a rational point P of X over V. Since t@ and trCr are characterized by the mapping ~1”~(PK)<~~-~, respectively clcr(Pk), to 1, it suffices to show that
Q2d,2b(Cl~e(PK)C2b-d) = ,~c’(P,)t2b-d. But that follows from property (2). 0 Assume that k is algebraically closed and that X is irreducible. By construction o!,b is a Galois equivariant map compatible with the Frobenius action. Since both sides have the same rank over B,,, it suffices to show that the morphism (u,b is injective. This follows from the fact that both the domain and the target satisfy Poincare duality, and the map o,b commutes with products and traces (by the above lemma). The proof that c&b induces an isomorphism on filtrations is now standard (see, [18, 31). By Lemma 2.2.1 extension of d!,b to BdR is compatible with the filtrations. One passes to the associated grading and reduces to showing that the induced map s’ab : c,(z) Q,
jy(XF,
Q,(b))
-4 $ C,(b + ljCZ
j) 8.~ H”-‘(X,,
$ii,,)l
1 E ‘,
is an isomorphism (this is the statement of the Hodge-Tate Conjecture). The dimensions of both sides being equal, it suffices toI show that i&b is injective. Since both the target and the domain of ?&, satisfy Poincare duality and &b is compatible withi products (Lemma 5.1. I), ANNALES
SCIENTIFIQUES
DE L’kOLE
NORMAL]?
SUFdRIEURE
680
W. NIZIOt
for i&b to have a left inverse, it is sufficient that it be compatible with traces. But, &d,ab is obtained from the composition of the map
a;d,2b: H2d(X,,
,(2b)) -+ Q @ proj lim,H,$(Xv/W(k),
JFbl/JFb+ll)
induced by grading the maps o&2b, and the natural isomorphism ggr : Q @ proj Sm,H,2,d(Xv/W(k),
Ji2b1/JFb+11) N 6,(26
118, 1.41
- d) @ Hd(XK,
fl$K,zv).
Since the Hodge trace is compatible with the de Rham trace, we can use Lemma 2.2.1, Lemma 5.1.3, and the compatibility of the Berthelot-Qgus isomorphism with traces [18, B.3.31 to conclude our H:,(Xo/W(~)) @w(rc) K - H&(X,/W proof. 0
REFERENCES [I]
Theorie de Dieudonne’ cristalline. II, Lect. Notes in Math., vol. 930, York, 1982. P. BERTHELOT, A. GROTHENDIECK and L. ILLUSIE, Theorie des intersections et theoreme de Riemann-Roth, Lect. Notes in Math., vol. 225, Springer-Verlag, Berlin, Heidelberg and New York, 1971. P. BERTHELOT and A. OGUS, F-isoctystals and de Rham cohomology. I, Inv. Math. 72, 1983, pp. 159-199. W. DWYER and E. FRIEDLANDER, Algebraic and &ale K-theory, Trans. Amer. Math. Sot. 292, 1985, no. 1, pp. 247-280. G. FALTINGS, Crystalline cohomology and p-adic Galois representations, Algebraic analysis, geometry and number theory (J. I. Igusa ed.), Johns Hopkins University Press, Baltimore, 1989, pp. 25-80. 6. FALTINGS, p-adic Hodge-theory, J. of the AMS 1, 1988, pp. 255-299. J.-M. FONTAINE, Cohomologie de de Rham, cohomologie crystalline et representations p-adiques, Algebraic Geometry Tokyo-Kyoto, Lect. Notes Math., vol. 1016, Springer-Verlag, Berlin, Heidelberg and New York, 1983, pp. 86-108. J.-M. FONTAINE, Sur certains types de representations p-adiques du groupe de Galois d’un corps local, construction d’un anneau de Barsotti-Tate, Ann. of Math. 115, 1982, pp. 529-577. J.-M. FONTAINE, Le corps des periodes p-adiques, A&risque, vol. 223, 1994, pp. 321-347. J.-M. FONTAINE and G. LAFAILLE, Construction de representations p-adiques, Ann. Sci. .Ec. Norm. Sup., IV. Ser., 15, 1982, pp. 547-608. J.-M. FONTAINE and W. MESSING, p-adic periods and p-adic &ale cohomology, Current Trends in Arithmetical Algebraic Geometry (K. Ribet, ed.), Contemporary Math., vol. 67, Amer. Math. Sot., Providence, 1987, pp. 179-207. E. FRIEDLANDER, l&ale K-theory. II. Connections with algebraic K-theory, Ann. Sci. Ecole Norm. Sup., IV. Ser., 15, 1982, pp. 231-256. H. GILLET, Riemann-Roth theorems for higher algebraic K-theory, Adv. Math. 40, 1981, pp. 203-289. H. GILLET and W. MESSING, Cycle classes and Riemann-Rock for crystalline cokomology, Duke Math. J., 55, 1987, pp. 501-538. M. GROS, Regulateurs syntomiques et valeurs defonctions Lp-adiques I, Invent. Math., 99, 1990, pp. 293-320. M. GROS, Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmic, MCm. Sot. Math. France (N.S.) 21, 1985. BZ. KATO, On p-adic vanishing cycles (application of ideas of Fontaine-Messing), Algebraic Geometry, Sendai, 1985, Adv. Stud. Pure Math. 10, North-Holland, Amsterdam-New York, 1987, pp. 207-251. K. KATO and W. MESSING, Syntomic cohomology and p-adic &tale cohomology, TBhoku Math. J. 44, 1992, pp. 1-9. D. QUILLEN, Higher algebraic K-theory I, Algebraic K-theory I, Lecture Notes in Math. 341, Springer-Verlag, Berlin-Heidelberg-New-York, 1973, pp. 85-147. V. V. SHEKHTMAN, Ckern classes in algebraic K-theory, Trans. Moscow Math. Sot., Issue I, 1984, pp. 243-271. P. BERTHELOT,
L. BREEN and W. MESSING,
Springer-Verlag,
[2] ;3] [4] [5] [6] [7]
[8] [9]
[lo] [I I]
[ 121 [13]
[14] [ 151 [16] [17] [18] [19] [ZO]
4e SeRIE - TOME 31 -
Berlin-New
1998 - No 5
CRYSTALLINE [21] C. [22] C. [23] C. [24] R. [25] R.
CONJECTURE
VIA
SOULS, Operations on &ale K-theory. Applications., Algebraic K-theory I, Lect. Notes Math. 966, Springer-Verlag, Berlin, Heidelberg and New York, 1982, pp. 271-303. SOUL& K-thkoorie des anneaux d’enti,ers de corps de nombres et cohomologie &ale, Inv. Math. 55, 1979, pp. 251-295. SOUL& Ope’rations en K-thkorie algibbrique, Canad. .I. Math. 37, no. 3, 1985, pp. 488-550. THOMASON, Algebraic K-theory and &ale cohomology, Ann. Scient. Ecole Norm. Sup. 18, 1985, pp. 431-552. THOMASON, Bott stability in algebraic K-theory, Applications of algebraic K-theory to algebraic geometry and number theory I, II (Boulder, Colo., 1983), Contemporary Math., vol. 55, Amer. Math. Sot., Providence, 1986, pp. 389-406.
(Manuscript received accepted after revision
W. N~z~ot Department of Mathematics, College of Science, University of Utah, Salt Lake Utah 84112-0090, USA SE-mail:
ANNALES
681
K-THEORY
City
[email protected]
SCIENTF'IQUES
DEL'kOLENORMALE
SUPrjRIEURE
October March
21, 1996; 20, 1998.)