WEIGHT HOMOLOGY OF MOTIVESSHANE KELLY AND SHUJI SAITO

Abstract. In the first half of this article we define a new weight homology
functor on Voevodsky’s category of effective motives, and investigate some ofits properties. In special cases we recover Gillet-Soulé’s weight homology, andGeisser’s Kato-Suslin homology. In the second half, we consider the notions of“co-étale” and “reduced” motives, and use the notions to a prove a theorem
comparing motivic homology to étale motivic homology.
Due to [Ke1] we do not have to restrict to smooth schemes.

Contents
1. Introduction
2. Weight homology of motives
3. Weight homology of schemes
4. Motivic homology and étale motivic homology
5. Étale cycle map for motivic homology
6. Reduced motives and co-étale motives
Appendix A. Étale realisations of non-smooth motives
References

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1. Introduction
Weight homology. In their highly cited work [GS1, §2], Gillet and Soulé associate
to every variety X over a perfect field1 k, a complex W (X) of (effective) Chow
motives, and show it is well-defined up to chain homotopy. Using this, in [GS1, §3]
they define the weight homology (with compact support) of X with (Γ-coefficients)
as
HnGS (X, Γ) = Hn (Γ(W (X)))
where Γ : Choweff → A is an additive covariant functor from effective Chow motives
Choweff to an abelian category A. A distinguished property of Gillet and Soulé’s
weight homology is : For X projective smooth over k, we haveΓ(X) for n = 0,
GS
Hn (X, Γ) =0
for n 6= 0.
On the other hand, for a variety X over a finite field k, Geisser defines in [Ge2, §8]a homology theory he calls the Kato-Suslin homology with coefficients in an abeliangroup A as
HnKS (X, A) = Hn ((C ∗ (X)(k) ⊗ A)G )where C ∗ (X)(k) = homCor (∆∗k , k×k X) and G is the Weil group of k (i.e., the subgroup of Gal(k/k) generated by the Frobenius). In [Ge2] Geisser shows the following
1In their article they state their result for fields of characteristic zero since they use resolution of

singularities, however if one uses Z[ p1 ]-coefficients (resp. Q-coefficients) one can replace this with theappropriate theorem of Gabber (resp. de Jong). Alternatively, one can work under the assumptionthat the field admits resolution of singularities.
1

2

SHANE KELLY AND SHUJI SAITO(see Prop. 3.4): Assume either that A is torsion prime to ch(k), or that Parshin’s
conjecture P0 stated in Prop. 3.4(2) holds. Then, for X smooth over k, we have c(X)
A
for n = 0KS
Hn (X, A) =0
for n 6= 0
where c(X) is the set of connected components of X.
The first main purpose of this article is to introduce a general homology theory
for motives from which the above two can be recovered as special cases.
Recall that in [Bo2] Bondarko extends Gillet-Soulé’s weight complex to a triangulated functor defined on all geometric effective motives (cf. Theorem 2.1). We
observe that a consequence of this is an induced equivalence of categories (see §2.1
for the notations):
Theorem (Thm. 2.4). Let k be a perfect field of exponential characteristic p. The
functor
additive functors
homological functors2(1.1)→
eff (k, Z[ 1 ])on DMgm
on Choweff (k, Z[ p1 ])
peff (k, Z[ 1 ]) induces an equivalence
induced by the inclusion Choweff (k, Z[ p1 ]) → DMgmpwhen restricted to the full subcategory of homological functors H satisfying:(SPA) H(M (X)[n]) = 0 when X is smooth and projective and n 6= 0.

For any additive covariant functor Γ : Choweff (k, Z[ p1 ]) → A we write
(1.2)

eff
H0W (−, Γ) : DMgm
(k, Z[ p1 ]) → A

for its associated homological functor (Def. 2.6) and write HnW (−, Γ) = H0W (−[n], Γ)for n ∈ Z. For an object A of the category AB of abelian groups, we put HnW (−, A) =HnW (−, ΓA ) where ΓA : Choweff (k, Z[ p1 ]) → AB is the additive functor such thatΓA (X) = Ac(X) for X smooth projective, where c(X) is the set of connected components of X.
In this way, we obtain a general weight homology theory. Our theory recovers theother two:
Theorem (Thm. 3.1, Thm. 3.6). Let k be a perfect field and Sch/k the category offinite type separated k-schemes.
(1) For an additive covariant functor Γ : Choweff (k, Z[ p1 ]) → A, there are canonical isomorphisms for X ∈ Sch/k
H GS (X, Γ) ∼= H W (M c (X), Γ)nnwhich are covariantly functorial for proper morphisms in Sch/k.
(2) Assume k is a finite field. If A is a torsion abelian group of torsion prime
to the characteristic, there are canonical isomorphisms for X ∈ Sch/k(1.3)
H KS (X, A) ∼= H W (M (X), A)nnwhich are covariantly functorial for morphisms in Sch/k. Moreover, for ageneral Z[1/p]-module A, the existence of isomorphisms such as (1.3) follows
from Parshin’s conjecture P0 .
See §3.1 and §3.2 for a number of properties that our weight homology (and consequently, Gillet-Soulé weight homology and Geisser’s Kato-Suslin homology) satisfies.
In particular, we notice that HnW (M c (−), Γ) is contravariant for quasi-finite flat morphisms in Sch/k, and for arbitrary morphisms in Sm/k between schemes of the same
dimension (our properties (Wc3) and (Wc4)). The corresponding contravariance of

WEIGHT HOMOLOGY OF MOTIVES

3

Gillet-Soulé’s theory HnGS (−, Γ) seems to be new, and not at all obvious from its
construction.
For functors Γ on Choweff (k, R) such as ΓA which are P1 -invariant, the associatedweight homology functor satisfies a number of additional interesting properties: itvanishes on all twisted motives, it is birationally invariant on smooth (not necessarily
projective) schemes, and the weight homology of (not necessarily projective) smooth
schemes is concentrated in degree zero (cf. Lem. 2.7 and Thm. 2.8).
Cycle maps. The canonical functoreff
α∗ : DM eff (k, R) → DMét(k, R)

connecting Voevodsky’s category of motives and its étale version admits a rightadjoint α∗ . The motivic homology and étale motivic homology with R-coefficientsof an object C in DM eff (k, R) are defined as
HiM (C, R) = homDM eff (k,R) (R[i], C),

and

HiM,ét (C, R) = homDM eff (k,R) (R[i], α∗ α∗ C)
where R[i] = M (k)[i] is a shift of the unit for the tensor structures. The second halfof the article concerns the following maps functorial in C ∈ DM eff (k, R):α∗ : HiM (C, R) → HiM,ét (C, R)
induced by the unit id → α∗ α∗ of the adjunction (α∗ , α∗ ).
Recall that motivic homology is related to fundamental invariants of schemes. For
example, if p is invertible in R, or if we assume strong resolution of singularities, for
X ∈ Sch/k, r, i ∈ Z, there are canonical isomorphisms ([V, §2.2] and [Ke1, Chap. 5])
(1.4)
(1.5)

HiM (M (X), R) ∼= HiS (X, R)
H M (M c (X)(r), R) ∼= CH−r (X, i + 2r; R)iwhere HiS (X, R) is the Suslin homology with R-coefficients ([SV1]), and CH∗ (X, •; R)
are the higher Chow groups with R-coefficients ([B]).
On the ther hand, for R = Z/nZ with n coprime to p, the étale motivic homology
M,ét
Hi(C, Z/nZ) for C = M (X) (resp. M c (X)) are identified with the dual of étale
cohomology (resp. étale cohomology with compact support) (see Prop. 5.4 andLem. 5.5).
The main theorem is:
Theorem (Thm. 4.2). Let p be the exponential characteristic of k, let n be an
integer coprime to p, and let R be Z/nZ or Z[1/p].
(1) If k is algebraically closed, α∗ is an isomorphism.
eff (k, R), there is a canonical long exact sequence(2) If k is finite and C ∈ DMgmα∗W
M
· · · → HiM (C, R) −→ HiM,ét (C, R) → Hi+1
(C, Λ) → Hi−1
(C, R) → · · ·which is functorial in C where
Q/Z[1/p]Λ=Z/nZ

if R = Z[1/p],
if R = Z/nZ.

In view of the remarks above the theorem, the first part of the theorem implies that
if k is algebraically closed, then there are canonical isomorphisms for X ∈ Sch/k:
HiS (X, Z/nZ) ∼= H i (Xét , Z/nZ)∗ , andCH0 (X, i; Z/nZ) ∼= H i (Xét , Z/nZ)∗ .
c

4

SHANE KELLY AND SHUJI SAITOOne can identify these isomorphisms with those in [SV1] and [Su]. (This identification is essentially a consequence of the Suslin homology/cohomology pairing and theétale homology/cohomology pairing being compatible. See [Ke2, Prop. 15 and 17]
for the details). Our proof of the above theorem is simpler than those in [SV1] and[Su], but depends on the Beilinson-Lichtenbaum conjecture due to Suslin-Voevodsky[SV3] and Geisser-Levine [GL, Thm.1.5] which relies on the Bloch-Kato conjecture
proved by Rost-Voevodsky (see the proof of Lem. 5.6).
For a finite field k and R = Λ = Z/nZ with n coprime to p, the map α∗ gives rise
to maps
HiS (X, Z/nZ) → H i+1 (Xét , Z/nZ)∗ ,CH0 (X, i; Z/nZ) → Hci+1 (Xét , Z/nZ)∗ ,and the second part of the above theorem relates the kernel and cokernel of thesemaps with weight homology and weight homology with compact support of X introduced in §3. In particular, if X is smooth over k, we get canonical isomorphisms
HiS (X, Z/nZ) ∼= H i+1 (Xét , Z/nZ)∗

for i ∈ Z.

See Cor. 4.3 for the details and Rem. 4.4 for relation of these results with [Ge2].
In order to show the second part of the above theorem, we introduce new homology
functors which seem to be of independent interest. The idea is to introduce an endofunctor Θ : DM eff (k, R) → DM eff (k, R) equipped with a natural transformation
Θ → id, which gives rise to distinguished triangles (Equation 6.2)
ΘC → C → α∗ α∗ C → ΘC[1]eff (k, R), allowing one to define the “co-étale” part ΘC of a motive C.
for C ∈ DMét
Then the second part of the above theorem is equivalent to the assertion that if keff (k, R):is finite, there are canonical isomorphisms for C ∈ DMgm

(1.6)

∼
=

W
HiM (Θ(C)), R) −→ Hi+1
(C, Λ)

for i ∈ Z,

which are natural in C (see Theorem 6.3). To show this, we use d≤0 DM eff (k, R),
the localizing subcategory of DM eff (k, R) generated by motives of smooth varietiesof dimension zero (cf. [V, §3.4]). By [A-BV] there is a projection functor Lπ0 :DM eff (k, R) → d≤0 DM eff (k, R), the “field of constants” part (cf. Equation (6.6)).
Similarly as above, we get again distinguished triangles (Equation 6.8)
Cred → C → Lπ0 (C) → Cred [1]eff (k, R), allowing one to define the “co-dimension-zero” or “reduced”
for C ∈ DMétpart Cred of a motive C. Surprisingly, if k is a finite field, HiM (Θ(Cred ), R) = 0 for
all i ∈ Z, or equivalently the canonical morphism

Θ(C) → Θ(Lπ0 (C))
induces isomorphisms HiM (Θ(C), R) ∼= HiM (Θ(Lπ0 (C)), R) for all i ∈ Z (Claim
6.5(2), Equation (6.9)). Heuristically, this can be interpreted as saying that over afinite field, all the difference between the motivic homology and the étale motivic
homology is contained in the field of constants. Finally the desired isomorphismseff (k, R):(1.6) are obtained by showing canonical isomorphisms for C ∈ DMgm
(1.7)

∼
=

W
HiM (Θ(Lπ0 (C)), R) −→ Hi+1
(C, Λ)

which are natural in C (see (6.10)).

for i ∈ Z,

WEIGHT HOMOLOGY OF MOTIVES

5

Outline. In Section 2 we present our notation and conventions, and then the definition of our weight homology of motives functor (Def. 2.6). We also discuss therelationship between some “birational” and “weight” properties homological functors might satisfy in Lemma 2.7. Our weight homology functor satisfies all of them
(Theorem 2.8).
In Section 3 we specialise to the case of weight homology of motives of schemes
and motives of schemes with compact support. It is in this section that we show that
our theory recovers the Gillet-Soulé weight homology over a perfect field (Thm. 3.1),
and Geisser’s Kato-Suslin homology over a finite field if the coefficient group is finite
torsion prime to the characteristic of the base field (Thm. 3.6).
In Section 4 we present our main comparison theorem between motivic and étale
motivic homology (Theorem 4.2) and give some consequences. The theorem isproved in the subsequent sections. Section 5 is the comparison in the case of motivesof smooth schemes with compact support (Theorem 5.1).
In Section 6 we present the notions of “co-étale” and “reduced” motives and use
them to prove the general case of Theorem 4.2 (see Theorem 6.3).
In Appendix A we identify that the étale realisation of motives of singular schemes(and motives with compact support of singular schemes) are what one expects them
to be.
Acknowledgements. We are indebted to B. Kahn for some ideas used in thispaper. In particular, the idea of using the reduced part of a motive to study thecycle map over a finite field is due to him. We also thank T. Geisser for helping us
to understand the relationship between our work and his.
Warning. We almost always work under the hypothesis that either the base fieldsatisfies resolution of singularities, or alternatively, that the coefficients (A, Λ, R,
Z/nZ, or Γ depending on the statement) are Z[ p1 ]-linear. We will remind the reader
from time to time.
2. Weight homology of motives
2.1. Notation and conventions. Through-out, k will be a perfect field of exponential characteristic p and n a positive integer prime to p. We will remind the reader
occasionally. For a ring R, we write DM eff (k, R) for the tensor triangulated category of (unbounded) effective motives constructed by Voevodsky [V]. We will use
Sm/k (resp. Sch/k) to denote the category of smooth (resp. separated finite type)k-schemes. If p is invertible in R, or if we assume strong resolution of singularities
the category DM eff (k, R) is equipped with functors(2.1)

M : Sch/k →DM eff (k, R) ;X 7→ M (X)(2.2)

M c : Schprop /k →DM eff (k, R) ;X 7→ M c (X)where Schprop /k is the subcategory of Sch/k with all objects but only proper morphisms [Ke1, Chap. 5] (or [V] for the version without conditions on the coefficients,
but which assumes resolution of singularities).
eff (k, R) of compact objects of DM eff (k, R).
We now focus on the subcategory DMgm
Let SmCor/k be Voevodsky’s category of smooth correspondances and K b (SmCor/k)be the homotopy category of bounded complexes in SmCor/k. Then there is a natural functor (cf. [V])

(2.3)

eff
π : K b (SmCor/k) → DMgm
(k, R)

such that M (X) = π(X[0]) for X ∈ Sm. We also consider the category of (covariant)effective Chow motives Choweff (k, R). One description of Choweff (k, R) is as the6

SHANE KELLY AND SHUJI SAITO

smallest idempotent complete full subcategory of DM eff (k, R) containing the objects
M (X) for all X smooth and projective ([Ke1, 5.5.11(4)] or [V, 4.2.6]). From thisdescription there is a tautological inclusioneffι : Choweff (k, R) → DMgm
(k, R).

(2.4)

2.2. Weight structures and weight homology. One of the major accomplishments of Bondarko’s theory of weight structures is the following which extends GilletSoulé’s weight complex of a variety to a weight complex for any motive (cf. [Bo2,
Prop. 6.6.2]).
Theorem 2.1 (Bondarko [Bo2]). Let k be a perfect field of exponential characteristic
p. There exists a functorefft : DMgm(k, Z[ p1 ]) → K b (Choweff (k, Z[ p1 ]))which is triangulated and such that the composition
ι

t

eff
(k, Z[ p1 ]) → K b (Choweff (k, Z[ p1 ]))Choweff (k, Z[ p1 ]) → DMgm

is the canonical inclusion of Choweff (k, Z[ p1 ]) into its bounded homotopy category ascomplexes concentrated in degree zero. The functor t is called the weight complex
functor.
Remark 2.2. Bondarko shows that if Q is used instead of Z[ p1 ], the functor t is anequivalence if and only if the Beilinson-Parshin conjecture holds ([Bo, §8.3.2]).
Definition 2.3. Let T be a triangulated category and A be an abelian category.
(1) A homological functor H : T → A is an additive functor (preserving all small
sums that exist in T ) such that for any distinguished triangle T1 → T2 →+T3 −→ T1 [1], the sequence· · · → H(T1 ) → H(T2 ) → H(T3 ) → H(T1 [1]) → H(T1 [1]) → · · ·is exact.
eff (k, R) → A is SmP roj-acyclic if:(2) A homological functor H : DMgm
(SPA) for every smooth projective X we have H(M (X)[i]) = 0 for i 6= 0.
Theorem 2.1 has the following consequence.
Theorem 2.4. Let k be a perfect field of exponential characteristic p. Composiιeff (k, Z[ 1 ]) induces an equivalence oftion with the inclusion Choweff (k, Z[ p1 ]) → DMgmpcategories SmP roj-acyclic
additive functors
homological functors
(2.5)
ι∗ :
→
Choweff (k, Z[ p1 ]) → A
 DM eff (k, Z[ 1 ]) → A gmp

Furthermore, this equivalence induces canonical isomorphisms


i
∼
(2.6)
H(M (V• )[i]) = H (ι∗ H)(V• )for an object H of the left category, where V• is a bounded complex of smooth proeff (k, Z[ 1 ]), andjective varieties in SmCor/k, by M (V• ) we mean its image in DMgmp(ι∗ H)(V• ) is the complex in A obtained by applying ι∗ H term-wise. These isomorphisms are natural in H and in V• ∈ Compb (SmP rojCor/k).

WEIGHT HOMOLOGY OF MOTIVES

7

Proof. ι∗ is essentially surjective: Every additive functor Γ : Choweff (k, Z[ p1 ]) → A
has a canonical extension to a homological functor K b (Choweff (k, Z[ p1 ])) → A, which
sends a complex C • to H 0 ΓC • , the zeroth cohomology of the complex of abeliangroups obtained by applying Γ term-wise to C • . Let ι∗ Γ = H 0 Γ ◦ t denote thecomposition of this extension with the weight complex functor. The association ι∗is clearly functorial in Γ. Furthermore, the functors ι∗ Γ are SmP roj-acyclic sincefor every smooth projective X, the complex tM (X) is concentrated in degree zero.
So ι∗ gives a right inverse to ι∗ by Theorem 2.1, hence ι∗ is essentially surjective.
The isomorphisms of Equation (2.6): Let SmP rojCor/k denote the full subcategory of SmCor/k whose objects are smooth projective varieties. Then we have acommutative square of functors(2.7)

SmP rojCor/kπ/ K b (SmP rojCor/k)ι

Choweff (k, Z[ p1 ])



ι

π

/ DM eff (k, Z[ 1 ])gmp

and the isomorphisms of Equation (2.6) are a consequence of Lemma 2.5 below
applied to B = SmP rojCor/k.
ι∗ is full faithful: Finally, we notice that the two vertical functors π in the squareof Equation (2.7) are full ([Ke1, 5.5.11(4)] or [V, 4.2.6]). Due to the lower row
being idempotent completions, the functors π are not essentially surjective, butevery object in their respective targets is a direct summand of an object in theirimages. So the association π∗ sending an additive functor Γ on Choweff (k, Z[ p1 ])eff (k, Z[ 1 ])) to π Γ = Γ ◦ π is fully faithful. Consequently, the fully(resp. DMgm∗pfaithfullness of the ι∗ of Theorem 2.4 follows from the fully faithfullness of the ι∗ ofLemma 2.5 (with A = SmP rojCor/k).
Lemma 2.5. Suppose that B is an additive category, and that H : K b (B) → A is a
B-acyclic homological functor, i.e., a homological functor satisfying the condition:(BA) H(B[i]) = 0 for any B ∈ B whenever i 6= 0.
Let ι∗ H denote the induced additive functor B → A. Then for every complex C • inK b (B) there are canonical isomorphisms


i
•
(2.8)
H(C • [i]) ∼
H
(ι
H)(C
)
=
∗
which are natural in H and C • , where and (ι∗ H)(C • ) is the complex of abelian groups
obtained by applying ι∗ H term-wise. Consequently, the assignment H 7→ ι∗ H is anequivalence from the category of B-acyclic homological functors K b (B) → A to thecategory of additive functors B → A.
Proof. This proof is by induction on the length of the complex. For complexes concentrated in one degree the isomorphisms (2.8) are equalities. Suppose we have theisomorphisms (2.8) for the subcategory of complexes of length < d and morphisms
concentrated in a range of degrees < d, and let C • be a complex concentrated indegrees a, . . . , a + d. There is a canonical distinguished triangleτ C • → C • → C a [−a] → τ C • [1]where τ C • is the same as C • in degrees a + 1, a + 2, . . . , a + d, and zero elsewhere.
Now for i 6= 0 we have H(C a [i]) = 0 and so the isomorphisms (2.8) for C • fori 6= a, a + 1 come from those for τ C • . For the case i = a, a + 1, notice that since C •8

SHANE KELLY AND SHUJI SAITO(resp. τ C • ) is zero in degree a − 1 (resp. a) we haveH a (ι∗ H)(C • ) = ker H(C a ) → H(C a+1 ) and




•
a+1
a+2
a+1
(ι∗ H)(τ C ) = ker H(C) → H(C) .
H
With this in mind, the long exact sequence of the above distinguished triangle together with the isomorphisms (2.8) of C a and τ C • then lead to the exact sequence
0 → H(C[a]) → H(C a ) → ker H(C a+1 )→H(C a+2 ) → H(C[a + 1]) → 0
from which we obtain the isomorphisms (2.8) for C • and i = a, a + 1. One can easily
check the naturality in H and C • of the isomorphisms constructed in this way. Definition 2.6. For an additive functor Γ : Choweff (k, Z[ p1 ]) → A, let
eff(k, Z[ p1 ]) → A
H0W (−, Γ) : DMgm

be the SmP roj-acyclic (Def. 2.3) homological functor associated to Γ by Theorem
2.4. We put
HiW (−, Γ) = H0W (−[i], Γ) for i ∈ Z.
For an abelian group A, we put
HiW (−, A) = HiW (−, ΓA ),where ΓA : Choweff (k, Z[ p1 ]) → AB is the additive functor such that ΓA (X) = Ac(X)
for X smooth projective, where c(X) is the set of connected components of X.
2.3. Additional properties of homological functors. We will observe a wideeff (k, R) satisfies stronger conditions than SmP rojclass of homological functors on DMgm
acyclicity (Def. 2.3) (see Theorem 2.8 below):
Lemma 2.7. Suppose either resolution of singularities holds, or R is a Z[ p1 ]-algebra.
eff (k, R) → A be a homological functor. The following three properties
Let H : DMgm
are equivalent.3(PI) The canonical morphism H(M (P1X )[i]) → H(M (X)[i]) is an isomorphism
for all smooth projective X, and all i ∈ Z.
eff (k, R) and i ∈ Z.
(TL) We have H(M (1)[i]) = 0 for all M ∈ DMgm(BI) for any dense open immersion U → X of smooth schemes the induced morphism H(M (U )[i]) → H(M (X)[i]) is an isomorphism for all i ∈ Z.
Moreover, a SmP roj-acyclic homological functor H satisfies the above equivalent
properties if and only if it is Sm-acyclic, i.e., if and only if:(SmA) For every smooth (not necessarily projective) X we have H(M (X)[i]) = 0
for i 6= 0.
eff (k, R)
Proof. (PI) ⇒ (TL). Since motives of smooth projective varieties generate DMgm
100
as a triangulated category it follows that H(M (P ) ⊗ M ) → H(M ) is an isomoreff (k, R). Now the isomorphism M (P1 ) = Z ⊕ Z(1)[2]
phism for all objects M 0 ∈ DMgm
implies the claim.
(TL) ⇒ (BI). This follows from the generalised Gysin distinguished triangle (forU → X a dense open immersion of smooth schemes of dimension d with closed
complement Z)

M (U ) → M (X) → M c (Z)∗ (d)[2d] → M (U )[1]3(PI) = P1 -Invariance, (TL) = Tate Local, (BI) = Birational Invariance.

WEIGHT HOMOLOGY OF MOTIVES

9([Ke1, Prop 5.5.5, Thm 5.5.14(3)], cf. [V, Prop. 4.1.5, Thm 4.3.7(3)]).
(BI) ⇒ (PI). This follows from a very neat geometric argument of Colliot-Thélène.
See [KaSu, Appendix A].
Now suppose that H is SmP roj-acyclic.
(BI) ⇒ (SmA). If resolution of singularities holds, this is a direct consequence ofthe existence of smooth compactifications. Otherwise: By Nagata’s compactification
theorem and Gabber’s alterations theorem [Il] for every smooth scheme U and every
l 6= p there is a roof of morphisms U ← U 0 → X 0 where X 0 is smooth and projective,
U 0 → X 0 is a dense open immersion, and U 0 → U is finite flat surjective of degree
prime to l. Since H is assumed to be SmP roj-acyclic, the property (BI) implies
that (SmA) is satisfied for the scheme U 0 . But M (−) is functorial in SmCor/k soby the trace formula, M (f )M (t f ) = deg f · idM (U ) , for the finite flat morphism f ,the motive M (U ) is a direct summand of M (U 0 ), at least Z(l) -linearly. But we were
free to choose l 6= p, and so M (U ) is a direct summand of M (U 0 ) as long as we
use Z[ p1 ]-linear coefficients (cf. [Ke1, Appendix A2]). Hence, Property (TL) for U 0
implies Property (TL) for U .
(Sm-acyclic) ⇒ (TL). Let X ∈ Sm/k. Since M (P1X ) ∼= M (X) ⊕ M (X)(1)[2],
the property (SmA) implies that H(M (X)(1)[i]) = 0 unless i = 2. But theisomorphism M (A1X − X) ∼= M (X) ⊕ M (X)(1)[1] with (SmA) then implies that
H(M (X)(1)[i]) = 0 unless i = 1. That is, H(M (X)(1)[i]) = 0 for all i. Since objects
eff (k, R) itof the for M (X) for X ∈ Sm/k generate the triangulated category DMgm
efffollows that H(M (1)[i]) = 0 for all M ∈ DMgm (k, R) and i ∈ Z.
Theorem 2.8. Suppose either resolution of singularities holds, or A is a Z[ p1 ]module. The homological functor H0W (−, A) from Definition 2.6 satisfies all properties of Lemma 2.7. That is, H0W (−, A) is Sm-acyclic.
Proof. Obviously ΓA (X) = ΓA (P1X ) for X smooth projective so that H0W (−, A)satisfies (PI). It also satisfies (N) by definition. Hence Theorem 2.8 follows from
Lemma 2.7.

3. Weight homology of schemes
As always, let k be a perfect field of exponential characteristic p. In this section
we fix an additive functor Γ : Choweff (k, Z[ p1 ]) → A.
3.1. Weight homology with compact support of schemes. For X ∈ Sch/kand i ∈ Z, we put
(3.1)

W
Hi,c
(X, Γ) = H0W (M c (X)[i], Γ).

Let Schprop /k be the category of the same objects as Sch/k but only proper morphisms. By [V, §2.2] (if resolution of singularities holds) or [Ke1], this gives rise tocovariant functors
WHi,c(−, Γ) : Schprop → A (i ∈ Z)
satisfying the following properties:(Wc1) (Nilpotent invariance) For X ∈ Sch/k with its reduced structure Xred ,→ X,
W (XW
the map Hi,cred , Γ) → Hi,c (X, Γ) is an isomorphism for any i ∈ Z.
(Wc2) (Localization axiom) For X ∈ Sch/k and a closed subscheme Z ⊂ X, one
has a functorial long exact sequence
W
W
W
W· · · → Hi,c
(Z, Γ) → Hi,c
(X, Γ) → Hi,c
(X − Z, Γ) → Hi−1,c(Z, Γ) · · ·10SHANE KELLY AND SHUJI SAITO(Wc3) (Pull-back by quasi-finite flat morphisms) For a quasi-finite flat morphism
f : Y → X in Sch/k, one has a canonical map
WW
f ∗ : Hi,c
(X, Γ) → Hi,c(Y, Γ)

compatible with functoriality for proper morphisms in Sch/k and with localization sequences in (Wc2) in an obvious sense.
(Wc4) (Pull-back by morphisms between smooth schemes) This is a consequence ofduality. For a morphism f : Y → X in Sm/k such that dim(Y ) = dim(X),one has a canonical map
WW
f ∗ : Hi,c
(X, Γ) → Hi,c(Y, Γ)

compatible with functoriality for proper morphisms in Sch/k and with localization sequences in (Wc2) in an obvious sense.
(Wc5) For X ∈ Sch/k projective smooth, we haveΓ(X) for i = 0
WHi,c (X, Γ) =0
for i 6= 0
(Wc6) If the groups Γ(X) are finitely generated for all X smooth and projective
W (X, Γ) are finitely generated for all(e.g., Γ = ΓZ/n ) then the groups Hi,c
X ∈ Sch/k.
Recall the definition of the Gillet-Soulé weight homology: For X ∈ Sch/k, let
X → X be an open immersion into a projective variety and Z the closed complement.
Take X • → X a proper cdh hypercover (or proper ldh-hypercover [Il], [Ke1]) suchthat each X i is smooth over k and let Z• → Z be a refinement of the induced
hypercover of Z such that each Zi is smooth. We then obtain a complex concentrated
in positive homological degrees in the additive envelope of the category of smoothprojective varieties equipped with an augmentationV• = Cone(Z• → X • ) → X.

(3.2)

By definition, the Gillet-Soulé weight complex W (X) is the image of V• in thehomotopy category of complexes of Chow motives K b (Choweff (k, Z[ p1 ])), and for an
additive functor Γ : Choweff (k, Z[ p1 ]) → AB one defines
HiGS (X, Γ) = Hi (Γ(V• )).

(3.3)

Theorem 3.1. There are isomorphisms
W
HiGS (X, Γ) ∼(X, Γ)= Hi,c(3.4)
natural in X ∈ Sch/k.

Remark 3.2. A remarkable point is that for HiGS (−, Γ), the properties (Wc3) and(Wc4) are not obvious from its construction.
Proof. The isomorphisms (3.4) come from Equation (2.6) and the morphism M (V• )→M c (X)eff (k, Z[ 1 ]) induced by Equation (3.2) which is an isomorphism by [Ke1, Thmin DMgmp5.3.1] (cf. [V, Thm 4.1.2]).

Example 3.3. Let E ∈ Sch/k be projective which is a simple normal crossing divisor
on a smooth P ∈ Sch/k. Let E1 , . . . , EN be the irreducible components of E andput for an integer a ≥ 0aE [a] =
Ei0 ,...,ia (Ei0 ,...,ia = Ei0 ∩ · · · ∩ Eia ).
1≤i0 <···<ia ≤N

WEIGHT HOMOLOGY OF MOTIVES

11

For Γ : Choweff (k, R) → A as above, consider a homological complex in A
∂a−1

∂

∂

1
a
Γ(E [0] ),Γ(E) : · · · → Γ(E [a] ) −→Γ(E [a−1] ) −→ · · · −→where Γ(E [a] ) is in degree a and ∂a is the alternating sum of the maps induced byinclusions Ei0 ,...,ia ,→ Ei0 ,...,ibν ,···a for ν = 0, . . . , a. Then, using (Wc2) and (Wc5),
one can show isomorphisms
WHa,c(E, Γ) ∼= Ha (Γ(E))

for a ∈ Z.

3.2. Weight homology of schemes. For X ∈ Sch/k and i ∈ Z, we put (cf.
Definition 2.6)
(3.5)

HiW (X, Γ) = HiW (M (X), Γ).

By [V, §2.2] (if resolution of singularities holds) or [Ke1] this gives rise to covariant
functors
HiW (−, Γ) : Sch/k → A (i ∈ Z)
satisfying the following properties:(W0) (Nilpotent invariance) For X ∈ Sch/k with its reduced structure Xred ,→ X,
the map HiW (Xred , Γ) → HiW (X, Γ) is an isomorphism for any i ∈ Z.
(W1) (Homotopy invariance) For X ∈ Sch/k, the map HiW (X×A1 , Γ) → HiW (X, Γ)
induced by the projection is an isomorphism for any i ∈ Z.
(W2) (Mayer-Vietoris axiom) For X ∈ Sch/k and an open covering X = U ∪ V ,one has a functorial long exact sequenceW· · · → HiW (U ∩V, Γ) → HiW (U, Γ)⊕HiW (V, Γ) → HiW (X, Γ) → Hi−1
(U ∩V, Γ) → · · ·(W3) (Abstract blow-up) Consider a cartesian diagram in Sch/k:p−1 (Z)

/ XZ




/X

Z

where pZ : XZ → Z is proper and Z → X is a closed immersion such that
p−1 (X − Z) → X − Z is an isomorphism. Then one has a functorial longexact sequence
−1
W
W
W
W
· · · → HiW (p−1Z (Z), Γ) → Hi (XZ , Γ)⊕Hi (Z, Γ) → Hi (X, Γ) → Hi−1 (pZ (Z), Γ) → · · ·(W4) For X ∈ Sm/k smooth projective, we haveΓ(X) for i = 0
W
Hi (X, Γ) =0
for i 6= 0
By Theorem 2.8, if Γ = ΓA for an abelian group A, we have the stronger property:(W5) For X ∈ Sm/k smooth, we have c(X)AW
Hi (X, A) =0

for i = 0
for i 6= 0

where c(X) is the set of connected components of X.
W (X, Γ) = H W (X, Γ).
(W6) For a proper X ∈ Sch/k, we have Hi,ci(W7) If the groups Γ(X) are finitely generated for all X smooth (e.g., Γ = ΓZ/n )then the groups HiW (X, Γ) are finitely generated for all X ∈ Sch/k if

12

SHANE KELLY AND SHUJI SAITO

In the rest of this section, we assume that k is a finite field with p = ch(k). Recall([Ge2, §8]) that since k is finite, one can define Geisser’s Kato-Suslin homology
HiKS (X, A) of X ∈ Sch/k with coefficients in an abelian group A as the ith homologyof the complex which we will denote by
id−ϕ X
KS
X
X(3.6)
RΓ (X, A) = Tot C∗ (k) ⊗ A → C∗ (k) ⊗ A → C∗ (k) ⊗ A
where k is the algebraic closure of the finite field k, the Frobenius of k is denotedby ϕ, and the presheaf of complexes C∗X (−) is the Suslin complex C ∗ (−) [SV1, §7]of the presheaf cequi (X/k) [SV2]. Consider the property (W5) for Geisser’s theory:(SmA)KS For X ∈ Sm/k smooth, we have c(X)
A
for i = 0KS
Hi (X, A) =0
for i 6= 0
where c(X) is the set of connected components of X.
Proposition 3.4 ([Ge2]).
(1) If A = Z/n with n coprime to p, then (SmA)KS is satisfied.
(2) If A is Z[1/p]-module, then (SmA)KS holds if the following condition holds:(P0 ) Rational Suslin homology is Sm-acyclic. That is, for all X smooth overa finite field, the rational Suslin homology groups HiS (X, Q) are zero for
i 6= 0.
Remark 3.5. (P0 ) is equivalent to the (part) of Parshin’s conjecture (see [Ge3]): For
X smooth and proper over k, CH0 (X, i) is torsion for i 6= 0.
Proof. Prop. 3.4(2) is Geisser’s [Ge2, Thm. 8.2]) (Indeed he shows also the converse
implication for A = Q). We give an alternative proof of Prop. 3.4. Notice that
RΓKS (X, A) is by definition the cone of the morphism
id−ϕ X
X
X
C∗ (k) ⊗ A → Tot C∗ (k) ⊗ A → C∗ (k) ⊗ A .
The ith homology of the source is by definition HiS (X, A). The ith homology of
the target is what Geisser defines to be the arithmetic homology and is denoted byar (X, A) ([Ge2, §7.1]). So now (SmA)
Hi+1
KS is equivalent to asking that the canonicalar (X, A) be isomorphisms for i 6= 0. By definition wemorphisms HiS (X, A) → Hi+1
have an exact sequence
(3.7)

S
ar
0 → Hi+1(X, A)G → Hi+1
(X, A) → HiS (X, A)G → 0,

where X = X ⊗k k with an algebraic closure k of k. By (1.4) and Th. 4.2 together
with (W5), we have isomorphisms for X smooth,
(3.8)

M,ét
HiS (X, A) ∼(M (X), A),= HiM (M (X), A) ∼
= Hi
M,ét
HiS (X, A) ∼(M (X), A).
= HiM (M (X), A) ∼= Hi

Letting Gk be the absolute Galois group of k, we have a spectral sequence
(3.9)

M,ét
2
Ei,j
= H −i (Gk , HjM,ét (M (X), A)) ⇒ Hi+j
(M (X), R),

Assuming either that A is torsion or that (P0 ) holds, HjM,ét (M (X), A) are torsion
for j 6= 0 and hence (3.9) gives rise toGM,ét
0 → Hi+1(M (X), A)G → HiM (M (X), A) → HiM,ét (M (X), A) → 0.

In view of (3.8), this provides an exact sequenceS
0 → Hi+1(X, A)G → HiS (X, A) → HiS (X, A)G → 0.

WEIGHT HOMOLOGY OF MOTIVES

13

ar (X, A).
Combined with (3.7), this proves the desired isomorphism HiS (X, A) ∼= Hi+1

Theorem 3.6. Assume k is finite and either A is a torsion prime to p or A is aZ[ p1 ]-module and (P0 ) holds. Then there are isomorphisms natural in X ∈ Sch/k:(3.10)HiKS (X, A) ∼= HiW (X, A),

where HiW (X, A) is the weight homology from §3.2.
Proof. Since the presheaves cequi (X/k) are functorial in X ∈ SmCor/k, the functor
RΓKS (−, A) is as well. In fact, using the total complex functor Tot, the functor
RΓKS (−, A) can be extended to a functor on the homotopy category K b (SmCor/k)of bounded complexes in SmCor/k, inducing a triangulated functor K b (SmCor/k) →K b (AB). For every X ∈ Sm/k and open subsets U, V ⊆ X, the functor RΓKS (−, A)
sends the complexes (A1X → X) and (U ∩ V → U ⊕ V → U ∪ V ) to acyclic complexes(−)(−)
since this is true for C∗ (k) ⊗ A and C∗ (k) ⊗ A. Hence RΓKS (−, A) factors via
a triangulated functor (cf. (2.3))
eff
RΓKS (−, A) : DMgm
(k, R) → K b (AB).

By the usual cdh/ldh descent arguments one can show a canonical isomorphism
HiKS (X, A) ∼= H −i (RΓKS (M (X), A))

for X ∈ Sch/k.

By Prop. 3.4 the homological functor RΓKS (−, A) is SmP roj-acyclic. Hence it now
follows from the equivalence of Theorem 2.5 that H −i RΓKS (−, A) is canonicallyeff (k, R).
isomorphic to the homological functor HiW (−, A) on DMgm
4. Motivic homology and étale motivic homology
eff (k, R)
We continue with k a perfect field of exponential characteristic p. We let DMéteffdenote the étale version of DM (k, R) (i.e., constructed with étale sheaves withtransfers instead of Nisnevich ones) and

(4.1)

eff
α∗ : DM eff (k, R) → DMét(k, R)

the canonical functor induced by étale sheafification. The functor α∗ admits a rightadjoint which is denoted by α∗ .
Lemma 4.1. The right adjoint α∗ to α∗ (Equation (4.1)) is fully faithful.
Proof. It suffices to show that the counit map α∗ α∗ C → C is an isomorphism foreff (k, R). We may assume C = F [0] for a single étale sheaf F . Then theC ∈ DMét
assertion is true for tautological reasons: Indeed, H 0 (α∗ F ) is just F where we only
remember that it is a Nisnevich sheaf so that α∗ H 0 (α∗ F ) → F is an isomorphism.
On the other hand, for q > 0, the étale sheaf associated to H q (α∗ F ) is 0, i.e.
α∗ H q (α∗ F ) = 0. This completes the proof of the lemma.
The motivic homology and étale motivic homology with R-coefficients of an object
C in DM eff (k, R) are defined as(4.2)

HiM (C, R) = homDM eff (k,R) (R[i], C),(4.3)

HiM,ét (C, R) = homDM eff (k,R) (R[i], α∗ α∗ C)

andwhere R[i] = M (k)[i] is a shift of the unit for the tensor structures.

14

SHANE KELLY AND SHUJI SAITOThe unit id → α∗ α∗ of the adjunction (α∗ , α∗ ) induces a natural map
α∗ : HiM (C, R) → HiM,ét (C, R)
which is functorial in C ∈ DM eff (k, R).
eff (k, R) we put
For C ∈ DMgm
HiW (C, Λ) = H0W (C[i], Λ),eff (k, R) → AB is the weight homology functor from Definiwhere H0W (−, Λ) : DMgm
tion 2.6 for A = Λ.

Theorem 4.2. Let k be a field of exponential characteristic p and let n be an
integer prime to p. Suppose either R = Λ = Z/n, or R = Z[1/p] and Λ = Q/Z[1/p].
(1) If k is algebraically closed, α∗ : HiM (C, R) → HiM,ét (C, R) is an isomorphism.
eff (k, R), there is a canonical long exact sequence(2) If k is finite and C ∈ DMgmα∗W
M
· · · → HiM (C, R) −→ HiM,ét (C, R) → Hi+1
(C, Λ) → Hi−1
(C, R) → · · ·which is functorial in C.
The proof of the theorem will be given in the following sections.
Take R = Λ = Z/nZ with (n, p) = 1. Then for C = M (X) (resp. M c (X)), thegroups HiM,ét (C, Z/nZ) are identified with the dual of étale cohomology (resp. étale
cohomology with compact support) by Proposition 5.4 and Lemma 5.5 below. Thus
(1.4) and (1.5) and Theorem 4.2(2) imply the following:
Corollary 4.3. Assume k is finite. For an integer n prime to p, there are longexact sequences functorial in X ∈ Sch/k:(4.4)

W· · · → HiS (X, Z/nZ) → H i+1 (Xét , Z/nZ)∗ → Hi+1
(X, Z/nZ)S→ Hi−1(X, Z/nZ) → · · · ,

(4.5)

W
· · · → CH0 (X, i; Z/nZ) → Hci+1 (Xét , Z/nZ)∗ → Hi+1,c(X, Z/nZ)→ CH0 (X, i − 1; Z/nZ) → · · · ,W (X, Z/nZ) are the weight homology and the weight
where HiW (X, Z/nZ) and Hi,chomology with compact support (cf. (3.5) and (3.1)). In particular, if X is smooth,(W5) implies canonical isomorphisms(4.6)

HiS (X, Z/nZ) ∼= H i+1 (Xét , Z/nZ)∗

for i ∈ Z.

Remark 4.4. Under the assumption of Cor. 4.3, Geisser [Ge2] shows an isomorphism(4.7)

Hiar (X, Z/nZ) ∼= H i (Xét , Z/nZ)∗ ,where Hiar (X, Z/nZ) is the arithmetic homology (see the proof of Prop. 3.4). Under(4.7) and the identification of Theorem 3.6, the long exact sequence (4.4) is identified
with the long exact sequence from [Ge2, §8]:
(4.8)

ar
KS
· · · → HiS (X, Z/nZ) → Hi+1
(X, Z/nZ) → Hi+1
(X, Z/nZ)S(X, Z/nZ) → · · ·→ Hi−1

As a consequence of the long exact sequence (4.4) and property (W7) we obtainan alternative proof of the prime-to-p part of [Ge2, Thm. 6.1]:

WEIGHT HOMOLOGY OF MOTIVES

15

Theorem 4.5 (cf. [Ge2, Thm. 6.1]). For k a finite field and n prime to thecharacteristic, the groups HiS (X, Z/nZ) are finitely generated for all i and all X ∈Sch/k.
Remark 4.6. We have a natural isomorphism
H 1 (Xét , Z/nZ)∗ ∼= π ab (X)/n,1

where π1ab (X) is the abelian fundamental group of X. Hence (4.4) gives an exact
sequence
(4.9)

ρ

X
H2W (X, Z/nZ) → H0S (X, Z/nZ) −→π1ab (X)/n → H1W (X, Z/nZ),

Functoriality implies that for a closed point x ∈ X, the composite of ρX withZ = H0S (X, Z) → H0S (X, Z/nZ)
sends 1 ∈ Z to the Frobenius element at x in π1ab (X)/n. Thus (4.9) and (4.6)
provide a generalisation of tame class field theory for smooth schemes over a finite
field studied in [ScSp]. A generalisation of [ScSp] in a different direction is studied
in [GeSc].
Example 4.7. Here is an example of computation of HiW (X, Λ) for X ∈ Sch/k.
Assume that there is a cartesian diagram:p−1 (Z)

/ XZ





/X
Z
where pZ : XZ → Z is proper and Z → X is a closed immersion such that XZ issmooth, p−1 (X − Z) → X − Z is an isomorphism and the reduced part E of p−1 (Z)is a simple normal crossing divisor on XZ . Assume additionally that Zred is propersmooth (e.g. dim(Z) = 0). Then, using (W0), (W3), (W4), (Wc1) and Example
3.3, we get an isomorphism:
W
Ha+1(X, Λ) ∼= Ha (ΓE (Λ)) for a ≥ 1.
H1W (X, Λ) ∼= Ker Λc(E) → Λc(Z) ⊕ Λc(XZ ) ,where under the same notation as Example 3.3, ΓE (Λ) is the complex· · · → Λc(E

[a] )

∂

a
−→
Λc(E

[a−1] ) ∂a−1∂

1−→ · · · −→
Λc(E

[0] )

.

5. Étale cycle map for motivic homology
In this section we prove the following theorem.
Theorem 5.1. Let X be smooth over k and n a positive integer invertible in k. For
an integer r ≥ 0, the mapα∗ : HiM (M c (X)(r), Z/nZ) → HiM,ét (M c (X)(r), Z/nZ)
is an isomorphism under one of the following conditions:(i) k is algebraically closed.
(ii) k is finite and r > 0.
(iii) k is finite, X is proper over k, and r = 0 and i 6= −1.
We need the following result:
Proposition 5.2. If k has finite cohomological dimension then:(1) If R is a Q-algebra, α∗ in Equation (4.1) is an equivalence of ⊗-triangulated
categories.
eff (k, R) = 0.
(2) If p = ch(k) > 0 and R is annhilated by a power of p, then DMét

16

SHANE KELLY AND SHUJI SAITO(3) For n prime to p, let Dét (k, Z/nZ) be the derived category of sheaves of
Z/nZ-modules over the small étale site Et(k) of Spec(k). Then the functoreffΦét : DMét(k, Z/nZ) → Dét (k, Z/nZ)

induced by the association F 7→ F(k̄) for an étale sheaf F on Sm/k, is anequivalence of ⊗-triangulated categories.
Proof. In the bounded above case, this is due to Voevodsky ([V, Prop. 3.3.2 and3.3.3]). The general case is shown by the same argument.
In the appendix we show that the functors Φét , α∗ , M, and M c are all compatible
in the way one would expect:
Proposition 5.3. Suppose that k is algebraically closed. For X ∈ Sch/k withstructural morphism π : X → Spec(k), there are canonical functorial isomorphisms
Rhom(Φét α∗ M (X), Z/nZ) = Rπ∗ (Z/nZ)X ,andRhom(Φét α∗ M c (X)), Z/nZ) = Rπ! (Z/nZ)X .
By Propositions 5.2 and A.2, we get the following.
Proposition 5.4. For X ∈ Sch/k and i, r ∈ Z, there are natural isomorphisms
H M,ét (M (X)(r), Z/nZ) ∼= H ét (X, Z/nZ(r)),
i
i
M,ét
c
ét
Hi (M (X)(r), Z/nZ) ∼(X, Z/nZ(r)).
= Hi,c

where
(5.1)

Hiét (X, Z/nZ(r)) = HomDét (k,Z/nZ) (Rπ∗ (Z/nZ)X , µ⊗rn [−i]),ét
Hi,c(X, Z/nZ(r)) = HomDét (k,Z/nZ) (Rπ! (Z/nZ)X , µ⊗rn [−i]).

called the étale homology and étale homology with compact support of X with
Z/nZ-coefficients respectively.
We now “convert” étale homology into cohomology:
Lemma 5.5. Let X ∈ Sch/k and let n be a positive integer invertible in k.
(1) If X is smooth over k of pure dimension d, there are canonical isomorphisms
H ét (X, Z/nZ(r)) ∼= H 2d−i (Xét , µ⊗d+r ),
i
c
n
ét
2d−i
Hi,c (X, Z/nZ(r)) ∼(Xét , µ⊗d+r).
=Hn(2) If k is algebraically closed there are canonical isomorphisms
H ét (X, Z/nZ(r)) ∼= H i (Xét , µ⊗−r )∗ ,
i
n
ét
i
∼
Hi,c (X, Z/nZ(r)) = Hc (Xét , µ⊗−r)∗ .
n(3) If k is finite there are canonical isomorphisms
H ét (X, Z/nZ(r)) ∼= H i+1 (Xét , µ⊗−r )∗ ,

i
n
ét
i+1
∼
Hi,c (X, Z/nZ(r)) = Hc (Xét , µ⊗−r)∗ .
nProof. By duality, we have natural isomorphisms
homDét (k,Z/nZ) (Rπ∗ (Z/nZ)X , Z/nZ) ∼= Rπ! Rπ ! Z/nZ,
hom(Rπ! (Z/nZ)X , Z/nZ) ∼= Rπ∗ Rπ ! Z/nZ.
Dét (k,Z/nZ)

where homDét (k,Z/nZ) denotes the inner Hom in Dét (k, Z/nZ). For X ∈ Sm/k ofpure dimension d, we have Rπ ! Z/nZ = µ⊗dn [2d] and (1) and (2) follow from this. If
k is finite, (3) follows from the Tate duality for Galois cohomology of finite fields. 

WEIGHT HOMOLOGY OF MOTIVES

17

5.1. Proof of Theorem 5.1. Let X be smooth of pure dimension d over a perfectfield k. It is shown in [Iv2] that the following diagram is commutative:
HiM (M c (X)(r), Z/nZ)α∗/ H M,ét (M c (X)(r), Z/nZ)i' (∗2)' (∗1)



H 2d−i (Xét , µ⊗d+r)∗
nCH−r (X, i + 2r; Z/nZ)


=

clX

4

CHd+r (X, i + 2r; Z/nZ),
where clX is the étale cycle map defined by Geisser-Levine [GL], and (∗1) comesfrom Equation (1.5) and (∗2) from Equation (5.4) and Lemma 5.5(1).
Similarly, when k is algebraically closed, in [Ke2] it is shown that (for r ≥ 0) thissame diagram is commutative but where now clX is Suslin’s isomorphism from [Su](cf. [Ke1, Thm. 5.6.1]).
Hence Theorem 5.1 follows from the Theorem 5.6 below.
Theorem 5.6. Under the assumption of Theorem 5.1, the Geisser-Levine cycle map
d+r,q
clX
: CHd+r (X, q; Z/nZ) → H 2(d+r)−q (Xét , Z/nZ(d + r))∗are isomorphisms for all q ∈ Z.
Proof. The case (ii) (resp. (iii)) is shown in [JS, Lem.6.2] (resp. [KeS1, Thm.
(9.3]). The case (i) is a consequence of [Su] and [Ke1, Theo. 5.6.1]. Here we givean alternative proof of (i) following the proof of [JS, Lem.6.2]. It is simpler but
depends on the Beilinson-Lichtenbaum conjecture due to Suslin-Voevodsky [SV3]and Geisser-Levine [GL, Thm.1.5] which relies on the Bloch-Kato conjecture provedby Rost-Voevodsky.
By the localization theorem for higher Chow groups ([B] and [L]), we have theniveau spectral sequenceM
CH 1
E a,b =
CHa+r (x, a + b; Z/nZ) ⇒ CHd+r (X, a + b; Z/nZ).
x∈Xa

By the purity for étale cohomology, we have the niveau spectral sequenceM2(d+r)−a−ba−b+2rét 1
E a,b =
Hét(x, Z/nZ(a + r)) ⇒ Hét
(X, Z/nZ(d + r)).
x∈Xad+r,a+b∞
The cyle map clX
preserves the induced filtrations and induces maps on Ea,b
compatible with the map on E 1 -terms induced by the cycle maps for x ∈ Xa :a−b+2rclxa+r,a+b : CHa+r (x, a + b; Z/nZ) → Hét(x, Z/nZ(a + r)).

By the Beilinson-Lichtenbaum conjecture, clxa+r,a+b is an isomorphism if b ≥ r. On
1
the other hand, we have CH E a,b = 0 for b < r since CHi (L, j; Z/nZ) = 0 for a field L
1 = 0 for b < 2r since cd(κ(x)) = a for x ∈ X
and integers i > j, and we have ét Ea,baby the assumption that k is algebraically closed. Noting r ≥ 0, these imply that thecycle maps induce an isomorphism of the spectral sequences and hence the desired
isomorphism of Theorem 5.6.

6. Reduced motives and co-étale motives
In case k is finite, the weight homology functor in Definition 2.6 is compared witha different homology functor which is defined as the homology of the “co-étale” partof motives (see Theorem 6.3(2)). To introduce this, we need some preliminaries. In
this section we still work over a perfect field k.

18

SHANE KELLY AND SHUJI SAITO

6.1. A triangulated lemma. We will apply the following lemma twice; in Equation (6.1) and Equation (6.7).
Lemma 6.1. Suppose that B : T → S is a fully faithful exact functor of triangulated
categories which admits a left adjoint A : S → T . Then there is a triangulated
endofunctor
C:S→S
with natural transformations c : C → idS and d : BA → C[1] such that for any
object X ∈ S,
ηX
cX
dX
C(X) −→X −→BA(X) −→
C(X)[1]is a distinguished triangle of S, where ηX is the unit map of the adjunction. If small
direct sums are representable in S, T and A, B commute with them, so does C.
Proof. Take X ∈ S and choose an object C ∈ S fitting in a distinguished triangle
η+1

X
C → X −→
BA(X) −→

We want to show that C is determined by X uniquely up to unique isomorphism.
For this it suffices to see that, for any Y ∈ S, one has HomS (C, BA(Y )) = 0.
By adjunction, we are reduced to showing that A(C) = 0, equivalently the map
A(ηX ) : A(X) → ABA(X) is an isomorphism. This holds since A(ηX ) is split by
the map εA(X) : ABA(X) → A(X), where ε is the counit of the adjunction, andthe latter is an isomorphism by the full faithfulness of B. The last statement is also
obvious.
The first application of Lemma 6.1 is to the adjunction
eff(k, R),(α∗ , α∗ ) : DM eff (k, R)  DMét

we get an endomorphism
Θ : DM eff (k, R) → DM eff (k, R) ;(6.1)

C → Θ(C)equipped with natural transformations α∗ α∗ [−1] → Θ → id such that+1

Θ(C) → C → α∗ α∗ C −→(6.2)

is a distinguished triangle for every C ∈ DM eff (k, R). One could call Θ(C) theco-étale part of C since homDM eff (k,R) (Θ(C), C 0 ) = 0 for all C 0 in the image ofeff (k, R) → DM eff (k, R). Notice that a consequence of Proposition 5.2(1)α∗ : DMétand (2) is that Θ(C) is necessarily torsion prime to ch(k):Lemma 6.2. For every C ∈ DM eff (k, R) the canonical morphism∼

hocolim hom(Z/n, Θ(C)) −→ Θ(C),

(6.3)

n

is an isomorphism where hom is the inner Hom in DM eff (k, R) and n ranges overall positive integers invertible in k.
Note also that
(6.4)

Θ(hom(Z/n, C)) ' hom(Z/n, Θ(C)).

In view of (6.2) Theorem 4.2 is a consequence of the following.
Theorem 6.3. Let C ∈ DM eff (k, R). Suppose either R = Λ = Z/n, or R = Z[1/p]and Λ = Q/Z[1/p].
(1) If k is algebraically closed, HiM (Θ(C)), R) = 0 for i ∈ Z.
eff (k, R):(2) If k is finite, there are canonical isomorphisms for C ∈ DMgm∼=

W
HiM (Θ(C)), R) −→ Hi+1
(C, Λ)

which are natural in C.

for i ∈ Z,

WEIGHT HOMOLOGY OF MOTIVES

19

6.2. Reduced motives. For r ≥ 0, let d≤r DM eff (k, R) be the localizing subcategory of DM eff (k, R) generated by motives of smooth varieties of dimension ≤ r (cf.
[V, §3.4]). We have a natural equivalence of categoriesd≤0 DM eff (k, R) ∼= D(AR ),where AR is the category of additive contravariant functors on the category of permutation R[Gk ]-modules [V, Prop. 3.4.1].
In [A-BV], Ayoub and Barbieri-Viale construct the left adjoint
Lπ0 : DM eff (k, R) → d≤0 DM eff (k, R)(6.5)

to the inclusion functor d≤0 DM eff (k, R) → DM eff (k, R). It satisfies(6.6)

Lπ0 (M (X)) = M (π0 (X))

for X ∈ Sm,

where π0 (X) is the “scheme of constants” of X, i.e.
aπ0 (X) =Spec(Li ),i∈I

where {Xi }i∈I are the connected components of X and Li is the algebraic closureof k in the function field of Xi . The functor Lπ0 commutes with small direct sums.
By Lemma 6.1 there is a triangulated endofunctor(6.7)(−)red : DM eff (k, R) → DM eff (k, R) ;C → Credιwith natural transformations Lπ0 [−1] → (−)red → id such thatι+1

Cred −→ C → Lπ0 (C) −→(6.8)

is a distinguished triangle for every C ∈ DM eff (k, R).
Definition 6.4. We call Cred the reduced part of C. We may say that C is reduced
if the canonical morphism Cred → C is an isomorphism, or equivalently if Lπ0 C = 0.
6.3. Proof of Theorem 6.3. We start with the following:
Claim 6.5. Let C ∈ DM eff (k, R).
(1) If k is algebraically closed, we have HiM (Θ(C), R) = 0 for i ∈ Z.
(2) If k is finite, we have HiM (Θ(Cred ), R) = 0 for i ∈ Z.
Proof. Reduction to R = Z/nZ and C = M (X) with X smooth and proper over k:
Since M (k) is a compact object of DM eff (k, R), the functor
HiM (−, R) : DM eff (k, R) → ABcommutes with infinite direct sums, hence with hocolimn . Therefore, in light of theisomorphisms∼∼

Θ(C) −→ hocolim hom(Z/n, Θ(C)) −→ hocolim Θ(hom(Z/n, C))nnof Lemma 6.3 and Equation 6.4 we may assume C is killed by some n and R = Z/nZ.
Reduction to C = M (X): Since α∗ , α∗ and Lπ0 commute
L with infinite direct
sums, the statement of the claim is true for a direct sum α∈A Cα if it is true for+1

all Cα . Moreover, if C 0 → C → C 00 −→ is an exact triangle in DM eff (k, Z/n) andthe statement of the claim is true for C 0 and C 00 , then it is true for C. By Gabber’s
refinement of de Jong’s theorem [Il], motives of smooth projective varieties are densein DM eff (k, Z/nZ) in the sense that the smallest triangulated subcategory containing their shifts and stable under infinite direct sums is equal to DM eff (k, Z/nZ) (see[Ke1, Prop.5.5.3]). Thus it suffices to show Claim 6.5 for R = Z/nZ and C = M (X),
where X is a smooth and proper over k.

20

SHANE KELLY AND SHUJI SAITOProof of Claim 6.5(1): Since we have reduced to the case C = M (X) with X
smooth proper over k and R = Z/nZ, Claim 6.5(1) follows from Theorem 5.1.
Proof of Claim 6.5(2): Now we prove the second assertion of the claim, where
k is finite and C = M (X) with X smooth proper over k. We have a commutative
diagram···/ H M (M (X)red , Z/nZ)i/ H M (M (X), Z/nZ)

α∗

i



/ H M (Lπ0 M (X), Z/nZ)
i

α∗



/ ···

α∗

· · · / HiM,ét (M (X)red , Z/nZ) / HiM,ét (M (X), Z/nZ) / HiM,ét (Lπ0 M (X), Z/nZ) / · · ·We want to show that α∗ on the left hand side is an isomorphism. By Theorem 5.1
and Equation (6.6), α∗ in the middle and on the right hand side are isomorphisms
for i 6= −1. By Equation (6.6) and Equation (1.5) we haveM
M
H−1
(M (X), Z/nZ) = H−1(Lπ0 M (X), Z/nZ) = 0.

Hence we are reduced to showing that the map
M,ét
M,ét
H−1
(M (X), Z/nZ) → H−1(Lπ0 M (X), Z/nZ)

is an isomorphism. By (5.4) and Lemma 5.5, this follows from the fact that thetrace map1
H 2d+1 (Xét , µ⊗dn ) → H (π0 (X)ét , Z/nZ)
is an isomorphism. This completes the proof of Claim 6.5.
It remains to show Theorem 6.3(2). Assume k is finite. By definition we have a
distinguished triangle+1Θ(Cred ) → Θ(C) → Θ(Lπ0 (C)) −→ .
Hence Claim 6.5(2) implies
HiM (Θ(C), R) ∼= HiM (Θ(Lπ0 (C))), R)(6.9)

hence we are reduced to showingW
HiM (Θ(Lπ0 (C))), R) ∼(C, Λ).
= Hi+1(6.10)

By Theorem 2.4 this follows from the following.
Claim 6.6. Suppose k is finite. For X ∈ Sm,
 c(X)ΛM
Hi (Θ(Lπ0 (M (X))), R) =0

for i = −1
for i 6= −1

where c(X) is the set of connected components of X.
Proof. Writing(6.11)

e iM (C, R) = HiM (Θ(Lπ0 (C))), R)H

eff
for C ∈ DMgm
(k, R),eff (k, R). Note that by Equations (6.3) andthis gives a homological functor on DMgm
eff(6.4), we have for C ∈ DM (k, R),(6.12)

Θ(C) = Θ(Ctor )with Ctor = hocolim hom(Z/n, C).
n

We may assume X is geometrically connected over a field L finite over k. By
Equation (6.6), we have Lπ0 (M (X)) = M (Spec(L)). We easily see M (Spec(L))tor =ΛL [−1], where
M (Spec(L)) ⊗ Q/Z[1/p] if R = Z[1/p],
ΛL =M (Spec(L))
if R = Z/nZ.

WEIGHT HOMOLOGY OF MOTIVES

21

Hence we get a distinguished triangle+1

Θ(Lπ0 (M (X))) → ΛL [−1] → α∗ α∗ ΛL [−1] −→which induces a long exact sequence∗αe M (M (X), R) → H M (Spec(L), R) −→··· → H
HiM,ét (Spec(L), R)i+1
ie iM (M (X), R) → · · · .
→H
M (Spec(L), R) =By Theorem 5.1, α∗ is an isomorphism for i 6= −1. For i = −1, we have H−1
0 while by Lemma 5.5 we have
H M,ét (Spec(L), R) ∼= H 1 (Spec(k)ét , Λ) ∼= Λ,
−1where the second isomorphism is the trace map in the Tate duality for Galois cohomology of finite fields. This completes the proof of Claim 6.6.

Appendix A. Étale realisations of non-smooth motives
We now show the compatibility of the functors
effα∗ : DM eff (k, R) → DMét(k, R)eff(k, Z/nZ) → Dét (k, Z/nZ)Φét : DMét

M : Sch/k →DM eff (k, R) ;X 7→ M (X)M c : Schprop /k →DM eff (k, R) ;X 7→ M c (X)from Equation (4.1), Proposition 5.2, and Equations (2.1) and (2.2) respectively.
We begin by recalling results of Suslin-Voevodsky that we will use. Let Z/nZ → I ∗be an injective resolution of the constant qfh-sheaf associated to Z/nZ in Shv qfh (Sch/k).
Recall that every qfh-sheaf has a canonical structure of transfers (Yoneda’s Lemma
together with [SV2, Thm. 4.2.12(1)]). So the two categories Shv qfh (Cor/k) andShv qfh (Sch/k) are canonically equivalent.
For a scheme X ∈ Sch/k let [X] denote its corresponding object in Cor/k aswell as the presheaf with transfers on Cor/k that it represents. The two naturaltransformations
η : id → C ∗ (−),
ε : Z/nZ → I ∗in Comp(Shv qfh (Cor/k)) together with the global sections functor
Γ(k, −) : Shv qfh (Cor/k) → AB
induce canonical natural transformations
ε

(A.1)

hom(C∗ ([X]), Z/nZ) → hom(C∗ ([X]), I ∗ (k))(A.2)

hom(C ∗ ([X]), I ∗ ) → hom(C∗ ([X]), I ∗ (k))(A.3)

Γ(k,−)

ηhom(C ∗ ([X]), I ∗ ) → hom([X], I ∗ ) ∼= I ∗ (X)of complexes of abelian groups.
Theorem A.1 (Suslin-Voevodsky). Suppose that the field k is algebraically closed,
n is invertible in k, and X ∈ Sch/k. The complex I ∗ and the sheaves I i are allétale acyclic. That is, one hasiaét (H i (I ∗ )(−)) = 0 and Hét(−, I j ) = 0

for all i 6= 0, j ≥ 0.

Moreover, the two morphisms (A.1), (A.2), and (A.3) of complexes of abeliangroups are all quasi-isomorphisms.
In fact, if [X] is replaced by a bounded complex in Compb (Cor/k) via the obvious
use of the total complex functor, they remain quasi-isomorphisms.

22

SHANE KELLY AND SHUJI SAITOProof. The assertion that I ∗ is acyclic is [SV1, Thm. 10.2]. The assertion that theI i are acyclic can be proven using verbatim the techniques of [V, Prop. 3.1.7] with
“Nisnevich” replaced by “étale”.
The claim that Γ(k, −) and η induce a quasi-isomorphisms is [SV1, Thm. 7.6] (see[Ge1] for a discussion on why the characteristic zero hypothesis in Suslin-Voevodsky’s
statement is not necessary). The morphism induced by ε is a quasi-isomorphism
because the complex C∗ ([X]) is a complex of free abelian groups, i.e, a complex ofprojective objects.
The usual spectral sequence arguments allow us to pass from a single [X] to abounded complex.
Proposition A.2. Suppose that the field k is algebraically closed and n is invertible
in k. For X ∈ Sch/k with the natural morphism π : X → Spec(k), there arecanonical functorial isomorphisms(A.4)

Rhom(Φét α∗ M (X), Z/nZ) = Rπ∗ (Z/nZ)X ,(A.5)

Rhom(Φét α∗ M c (X)), Z/nZ) = Rπ! (Z/nZ)X .

and

Proof. We begin with the non-compact support case. Recall that by definition, themotive of a scheme X ∈ Sch/k is the image of the complex of presheaves witheff (k) is just thetransfers C ∗ ([X]) in DM eff (k). The image of this complex in DMététale sheafification ([V2, Cor. 5.29]) and since k is separably closed, the image ofthis in Dét (k, Z/nZ) is just its evaluation at k. Since the terms of this complex
are free abelian groups, they are projective objects of Shv ét (Et/k) ∼= AB and soRhom(Φét α∗ M (X), Z/nZ) is represented by the complex hom(C∗ ([X]), Z/nZ).
On the other hand, the complex Rπ∗ (Z/nZ)X is represented by I ∗ (X) where I ∗is any complex of sheaves on (Sch/k)ét for which each I i is an acyclic étale sheaf
and which is quasi-isomorphic to Z/nZ as a complex of étale sheaves. For example,
the I ∗ which we chose above serves this purpose by Theorem A.1.
So the zig-zag of quasi-isomorphismsε Γ(k,−) ηhom(C∗ ([X]), Z/nZ) → ← → I ∗ (X)of Theorem A.1 provides the isomorphism in Dét (k, Z/nZ) we are looking for.
Now let Schcomp (k) be the category whose objects are closed immersions Z → X
between proper schemes and morphisms are cartesian squares. There is a canonical functor towards Schprop (k) (the category with the same objects as Sch(k) but
only proper morphisms) which sends a closed immersion Z → X to its open complement X − Z. The functor Schcomp (k) → Schprop (k) is full and essentiallysurjective by Nagata’s compactification theorem. Furthermore, by the localisation distinguished triangle ([Ke1, Prop. 5.5.5] or [V, Prop. 4.1.5]) the functor
M : Schcomp (k) → DM eff (k) (which sends a closed immersion Z → X to the imageof the complex Tot(C ∗ ([Z]) → C ∗ ([X]))) factors through M c : Schprop /k → DM eff (k)(via the obvious canonical natural transformation induced by the short exact sequence 0 → [Z] → [X] → zequi (X/k, 0)). That is, M c (X) is represented by the complex C ∗ ([X c ]) where we define [X c ] to be the two term complex (. . . →0→[Z]→[X]→0→ . . . ).
On the étale side, the situation is the same. The object Rπ! (Z/nZ)X is (bydefinition) represented by the complex Tot(I ∗ (X)→I ∗ (Z)) where I ∗ is as above.
So the isomorphism of Equation (A.5) can be obtained from the zig-zag of quasiisomorphismsε Γ(k,−) ηhom(C∗ ([X c ]), Z/nZ) → ← → I ∗ (X c ).



WEIGHT HOMOLOGY OF MOTIVES

23

References[A-BV]
[B]
[Bo]

[Bo2]

[Ge1]
[Ge2]
[Ge3]
[GeSc]
[GS1]
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[GL]
[Il]
[Iv1]
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[J]
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[KaSu][Ke1][Ke2][KeS1][KeS2][L][Pir][ScSp]
[Su]
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[SV2]

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[V2]

SHANE KELLY AND SHUJI SAITO

V. Voevodsky, Cohomological theory of presheaves with transfers, in: Cycles, Transfer,and Motivic Homology Theories, Annals of Math. Studies 143, Princeton University
Press, 2000.