Arithmetic Algebraic Geometry
RIMS, September 11 – 15, 2006

Finiteness results for motivic cohomology
Shuji Saito (University of Tokyo)
joint work with Uwe Jannsen (University of Regensburg)

September 11, 2006

Let X be a regular scheme of finite type over Fp or Z.
There is the following ‘folklore conjecture’, generalizing theanalogous conjecture of Bass on K -groups:Conjecture Aq
HM(X , Z(r )) is finitely generated for all q, r .

HereqqHM(X , Z(r )) = HZar
(X , Z(r )M ) = CH r (X , 2r − q)is the motivic cohomology of X , defined as the Zariski
cohomology of the motivic complex Z(r )M or, equivalently, as
the Bloch’s higher Chow group.

A quick review on Bloch’s higher Chow group :z r (X , q) =free abelian group on the set of closed integral
subschemes of codimension r of X × ∆q
which intersects all faces ∆s ⊂ ∆q properly,
whereqX∆ = Spec(Z[t0 , . . . , tq ]/(ti − 1))
qi=0

is the algebraic standard q-simplex and faces ∆s ⊂ ∆q aregiven by substituting 0 for some of t0 , . . . , tq .
One can view ∆q as an algebraic analogue of the standardq-simplex in topology and z r (X , q) as an analogue of the freeabelian group on the set of continuous maps ∆q → X .

We then get Bloch’s cycle complex :∂∂z r (X , •) : · · · z r (X , q) −→ · · · z r (X , 1) −→ z r (X , 0),
where the boundary map ∂ is the alternating sum of pull-backmaps to faces of codimension 1.
Bloch’s higher Chow group is defined as :CHr (X , q) = Hq (z r (X , •)).
We noteCHr (X , 0) = CHr (X ),the Chow group of algebraic cycles of codimension r on X
modulo rational equivalence.
Remark z r (X , q) is a free abelian group of infinite rank.

The only known general results on Conjecture A are:
Theorem(Dirichlet, Mordell-Weil) The groups
2
HM(X , Z(1)) = Pic(X ),1HM(X , Z(1)) = H 0 (X , OX× )are finitely generated for X of arbitrary dimension (e.g. for
X = Spec(Ok ) with a number field k, LHS is its ideal classgroup and RHS is its unit group).
Theorem(Quillen) Conjecture A holds for X of dimension 1.
Theorem (Bloch, Kato-S., Colliot-Thélène-Sansuc-Soulé)If X is regular and projective of dimension d over Fp or Z,2dHM(X , Z(d)) = CHd (X ) = CH0 (X )is finitely generated.

Observation: All these results are obtained by consideringsome regulator maps or cycle maps.
E. g., the last result comes from higher class field theory: The
following reciprocity map is shown to be injective:
ρX : CHd (X ) = CH0 (X ) −−−→ π1ab (X ),X : a regular proper scheme of dimension d over Fp or Z,
π1ab (X ) : the abelian fundamental group of X .
In case X = Spec(Ok ) for a number field k, this is the Hilbertclass field theory.
Finiteness result on π1ab (X ) due to Katz-Lang then implies
that CH d (X ) is finitely generated.

Second observation: By a commutative diagram
CHd (X )/nρX−−−→π1ab (X )/n'y

ρ2d,d2dHM(X , Z/nZ(d)) −−X−→ Hét2d (X , Z/nZ(d)) ,where d = dim(X ),ρX is interpreted as a cycle map into étale cohomology. HereqHM(X , Z/nZ(r )) = CHr (X , 2r − q; Z/nZ)

is the motivic cohomology with finite coefficients defined byCHr (X , q; Z/nZ) = Hq (z r (X , •) ⊗L Z/nZ).

We have a short exact sequence:qq(X , Z/nZ(r ))(X , Z(r ))/n → HM
0 → HM
q+1(X , Z(r ))[n] → 0,→ HMn

where M[n] = Ker(M −→ M) for a module M.
We are led to the following weak form of Conjecture A :qConjecture B HM
(X , Z/nZ(r )) is finite for any integern > 0 (for X regular of finite type over Fp or Z).

Conjecture B relates to Conjecture A like the weakMordell-Weil theorem relates to the strong one.

Now we discuss the target of the cycle map and our results.
The definition of the suitable étale cohomology needs somecare, because we have to consider p-torsion in characteristic p.
There are two cases.
(geometric case) : X is a smooth scheme over Spec(F) where
F is a finite field,(arithmetic case) : X is a regular scheme flat of finite typeover Spec(Z).
First consider the geometric case.

Let X be a smooth variety over a finite field F.
Geisser and Levine constructed an étale cycle map:ss
ρs,rX : HM (X , Z/nZ(r )) → Hét (X , Z/nZ(r )),where Z/nZ(r ) on RHS denotes the following complex of étalesheaves on X :rZ/nZ(r ) = µ⊗rm ⊕ Wν ΩX ,log [−r ],if n = mp ν with p = ch(F) and m invertible in F. Hereµm = the étale sheaf of m-th roots of unity, andWν ΩrX ,log = logarithmic part of de Rham-Witt sheaf Wν ΩrX .

Theorem (M. Artin, Milne) Héts (X , Z/nZ(r )) is finite.
Thus the injectivity ofssρs,rX : HM (X , Z/nZ(r )) → Hét (X , Z/nZ(r )),s
would imply that HM(X , Z/nZ(r )) is finite.

Higher class field theory implies ρs,dX for d = dim(X ), induces'sHM(X , Z/nZ(d)) −→ Héts (X , Z/nZ(d)) for s = 2d.

We also remarksHM(X , Z/nZ(d)) = CHd (X , 2d − s; Z/nZ) = 0 for s > 2d.

Theorem 1 (Jannsen - S.)
Let X be smooth projective of dimension d over F.
Let q ≥ 0 be an integer and ` be a prime.
Then ρs,dX induces an isomorphism'sHM(X , Z/`n Z(d)) −→ Héts (X , Z/`n Z(d))
for 2d ≥ s ≥ 2d − q,if (BK)q+2,` holds over X , and one of the following holds :(1) q ≤ 2, or(2) (RS)d , or(3) (RES)q .

We explain and discuss the assumptions:(BK)q,` over X :The cup product for Galois cohomologyq1(L, Z/`Z(q))
HGal(L, Z/`Z(1))⊗q → HGal

is surjective for any field L finitely generated over a residue
field of X (Milnor-Bloch-Kato conjecture).
The surjectivity of the above map is known to hold if
q = 2 (Merkurjev-Suslin) or` = ch(L)(= ch(F)) (Bloch-Gabber-Kato) or` = 2 (Voevodsky).
It has been announced by Rost and Voevodsky that it holds ingeneral.

(RS)d : For any Y integral and proper of dimension ≤ d overF, there exists a proper birational morphism π : Y 0 → Y suchthat Y 0 is smooth over F.
For any U smooth of dimension ≤ d over F, there is an openimmersion U ,→ Y such that Y is projective smooth over Fand Y − U is a simple normal crossing divisor on Y .

(RES)q : ForZ , a regular scheme of fintie type over F,
D ⊂ Z , a simple normal crossing divisor on Z ,Y ⊂ Z , an integral closed subscheme of dimension ≤ q suchthat Y \D is regular,
there exists a projective morphism π : Z 0 → Z such that(1) Z 0 is regular and π −1 (Z − D) ' Z − D,(2) D 0 = π −1 (D) is a simple normal crossing divisor on Z 0 ,(3) the proper transform Y 0 of Y in X 0 is regular andnormally crossing to D 0 .

Theorem 1 in case q ≤ 2 follows from that under theassumption (RES)q , in view of the following:
Theorem R (Hironaka, Cossart, Cossart-Jannsen-S.)(RES)q holds for q ≤ 2.
The proof is based on the theory of characteristic polyhedra
developed in 1960’s by Hironaka.
A method of Zariski (refined later by Lipman) shows that any
excellent surface can be desingularized by a succession ofblowups and normalizations.
In the above theorem only blowups with a regular centernormally crossing to boundaries (= inverse images of D) areallowed in the process of desingularization.

Using a seminal result of Soulé, further generalized by Geisser,
Kahn, and Jannsen, Theorem 1 implies the following:
Corollary 1 Let the assumption be as in Theorem 1. Assume
further that X is finite-dimensional in the sense of Kimura andO’Sullivan. Then we haveMsHM(X , Z(d)) 'Héts (X , Z` (d))
all prime `for 2d − 1 ≥ s ≥ 2d − q,
if q ≤ 2, or (RS)d holds, or (RES)q holds.
Remark : It is conjectured that any smooth projective vairiety
X over a finite field is finite-dimensional. The conjecture istrue in case there exists a surjective finite map Y → X where
Y is a product of abelian varieties and curves.

Now we consider the arithmetic case.
Here we assume :X : regular and flat of finite type over S = Spec(Ok ),Ok : the ring of integers in k,k : a number field or its completion at a finite place.
We fix a prime p > 2.
We assume that all reduced closed fibers of X −→ S are
SNCD and that the fibers over every point of Spec(Ok ) ofcharacteristic p is reduced.

Then, by work of Geisser and K. Sato, there is a cycle mapsnsn
ρs,rX : HM (X , Z/p Z(r )) → Hét (X , Z/p Z(r )),where Z/p n Z(r ) is an incarnation of the étale motivic complexét⊗L Z/p n ZZ(r )M

constructed by Sato.
Theorem (Sato) Héts (X , Z/p n Z(r )) is finite.
(The proof depends on results of Bloch-Kato, Hyodo and Tsuji
on p-adic vanishing cycles.)sThus the injectivity of ρrX,s would implies HM(X , Z/p n Z(r )) isfinite.

Theorem 2 (Jannsen - S.) Let the assumption be as aboveand let d = dim(X ).
(1) Assume k is a local field and that p is different from theresidue characteristic. Assuming (BK)4,p over X ,'sHM(X , Z/p n Z(d)) −→ Héts (X , Z/p n Z(d)) for 2d ≥ s ≥ 2d−2.

(2) Assume k is a number field and that X has goodreduction over every point of Spec(Ok ) of characteristic
p. Assuming (BK)3,p over X ,'sHM(X , Z/p n Z(d)) −→ Héts (X , Z/p n Z(d)) for 2d ≥ s ≥ 2d−1.

Here Theorem 1 (geometic case) has been only partiallyextended to arithmetic case due to a technical obstacle inp-adic Hodge theory.

We explain the basic idea of the proof of the above results.
First look at the case that dim(X ) = 1, i.e.,
X is a proper smooth curve over a finite field F, orX = Spec(Ok ) for a number field k.
We have the localization sequence
MH 1 (η, Z/nZ(1)) →Hx2 (X , Z/nZ(1)) → H 2 (X , Z/nZ(1))
x∈X0→ H 2 (η, Z/nZ(1)) →M

Hx3 (X , Z/nZ(1)) → 0 ,x∈X0η : the generic point of X ,X0 : the set of the closed points of X .

Using the Kummer theory and the purity theorem for étale
cohomology, it induces an exact sequence
ρ2,1X
0 → CH1 (X )/n →Hét2 (X , Z/nZ(1)) →ιKer Br (k) →M
Br (kx ) ,x∈X0

k is the residue field of η (= the function field of X ),kx is the completion of k at x, andBr (k) (resp. Br (kx )) is the Brauer group of k (resp. kx ).
By a fundamental result in number theory (Hasse principle forcentral simple algebras over k), the last term vanishes, whichshows that ρX2,1 is an isomorphism.

In order to extend the above localization argument to thehigher dimensional case, we introduceKC• (X , Z/nZ) : (Kato complex for X of finite type over Z)
···

Ma+1
(κ(x), Z/nZ(a)) →HGalx∈Xa··· →MaHGal(κ(x), Z/nZ(a−1)) → ·x∈Xa−1M

2(κ(x), Z/nZ(1)) →HGalx∈X1M1(κ(x), Z/nZ),
HGal

x∈X0

Xa = {x ∈ X | dim {x} = a}, andthe term in degree a is the direct sum of Galois cohomology ofresidue fields κ(x) of x ∈ Xa .

In case X is a proper smooth curve over a finite field F or
X = Spec(Ok ) for a number field k,KC• (X , Z/nZ) =MM
2
1
HGal(κ(x), Z/nZ(1)) →HGal(κ(x), Z/nZ),x∈X1x∈X0

which is identified withιBr (k)[n] →MBr (kx )[n],x∈X0wheren

M[n] = Ker(M −→ M) for a module M.

We define the Kato homology of X byKHa (X , Z/nZ) = Ha (KC• (X , Z/nZ)) (a ≥ 0).
Lemma 1 Let d = dim(X ). Assuming (BK)q+1,` over X , wehave a long exact sequence
KHq+2 (X , Z/`n ) →ρ2d−q,d2d−q
HM(X , Z/`n (d)) X→ Hét2d−q (X , Z/`n (d))→ KHq+1 (X , Z/`n )In conclusion the bijectivity of ρ∗,dX for X regular proper ofdimension d over F or Z follows from the following conjecture:Conjecture K (Kato) Let X be eithera smooth proper variety over Spec(F), ora regular proper flat scheme over Spec(Ok ) for the ring Ok of
integers in a number field k.
In the latter case assume that either n is odd or k is totally
imaginary.
Then:KHa (X , Z/nZ) 'Z/nZ
0a=0a 6= 0 .

The case dim(X ) = 1 rephrases the classical result on theBrauer group of a global field.
Kato proved the conjecture in case dim(X ) = 2.
In general the following result has been known due toColliot-Thélène and Suwa (geometric case) and Jannsen-S.
(arithmetic case)
Theorem Let the assumption be as above. Then
Q/Z a = 0
KHa (X , Q/Z) '00<a≤3
whereKHa (X , Q/Z) = lim KHa (X , Z/nZ).
−→nTheorem 1 follows from the following result.
Theorem 3 (Jannsen-S.):Let X be projective smooth of dimension d over F.
Let t ≥ 1 be an integer.
Then we have
KHa (X , Q/Z) 'Q/Z0

a=00<a≤tif either t ≤ 4 or (RS)d , or (RES)t−2 holds.
The same conlusion holds by replacing Q/Z by Z/`n Z
assuming (BK)t,` .

We now discuss the proof of Theorem 3.
Define étale homology of separated schemes of finite typef : X → Spec(F) by−a
(X , Rf ! Z/nZ) .
Haét (X , Z/nZ) := Hét

Using purity results for étale cohomology, a method ofBloch-Ogus gives rise to the niveau spectral sequence
1Ep,q(X , Z/nZ) =M

p−q
HGal(κ(x), Z/nZ(p))

x∈Xpét⇒ Hp+q(X , Z/nZ) .

Theorem (Jannsen-S.-Sato): There is an isomophism ofcomplexes
1(X , Z/nZ) ∼E∗,−1= KC∗ (X , Z/nZ)(−) ,where C (−) means to take the negative differentials in acomplex C . Hence
2(X , Z/nZ).
KHa (X , Z/nZ) ∼= Ea,−1

Theorem 3 on the Kato conjecture now follows by studying
the niveau spectral sequence carefully.

The first key observation is that the Kato conjecture implies
the following fact : For
X , projective smooth over F,
Y = Y1 ∪ Y2 ∪ · · · ∪ YN ⊂ X , SNCD,KHa (U, Z/nZ) of U = X − Y has a combinatoric description
as the homology of the complex(Z/nZ)π0 (Y(d) )→ (Z/nZ)π0 (Y(d−1) )→. . . → (Z/nZ)π0 (Y(1) )→ (Z/nZ)π0 (X ) ,π0 (Y (a) ) is the set of the connected components of the sum of
all a-fold intersections of the irreducible components
Y1 , . . . , YN of Y .

Conversely the Kato conjecture for X is deduced from such acombinatoric description of KHa (U, Z/nZ) for a suitablechoice of U = X − Y .
On the other hand,a weight argument using the affine Lefschetz theorem and theWeil conjecture proved by Deligne shows :if one of the divisors Y1 , . . . , YN on X is very ample, the étale
homology Haét (U, Q/Z) for a ≤ d has the same combinatoricdescription.
(Here we need take the coefficient Q/Z, not Z/nZ since we
use a weight arguemnt.)

Since the niveau spectral sequence1(U, Z/nZ) =Ep,qM

p−q
HGal
(κ(x), Z/nZ(p))

x∈Upét⇒ Hp+q
(U, Z/nZ)satisfies
2(U, Q/Z) ' KHa (U, Q/Z),
Ea,−1
1Ea,q(U, Q/Z) = 0 for q < −1,

the above combinatoric desription of KHa (U, Q/Z) is deduced
from the vanishing of the higher terms of the niveau spectral
sequence for such U :∞Ea,q(U, Q/Z) = 0for q > −1.

In order to show the vanishing, we pick up any element∞α ∈ Ea,q(U, Q/Z)

and then take a hypersurface section of high degree
Z ⊂ X containg the support Supp(α) of α
so that α is killed under the restriction∞∞Ea,q(U, Q/Z) → Ea,q(U\Z , Q/Z).

The point is that the assumption (RES)q allows us to make avery careful choice of Z (after desingularizing Supp(α)) suchthat the above restriction map is shown to be injective by theinduction on dim(U). This implies α = 0. Q.E.D.