Mixed motives and algebraic cycles II

Masaki Hanamura

The idea of motives goes back to A. Grothendieck (cf. [Kl 1, 2] [Ma]). As he worked on the Weil conjecture on the zeta function of a variety X over a finite field k, he noticed the importance of studying the action of the Frobenius map on the ℓ-adic etale cohomology H^{i}(X ⊗_{k} k̄, ℚ_{ℓ}); to prove the conjecture, one must show the characteristic polynomial of the Frobenius is a ℚ-polynomial independent of ℓ, and its eigenvalues have expected absolute values. He postulated certain conjectures on algebraic cycle classes, called the standard conjectures, and observed that they imply the Weil conjecture.

The standard conjectures are concerned not only with ℓ-adic cohomology, but with any other Weil cohomology (such as Betti (singular) or de Rham cohomology) as well, and the consequences are further-reaching. They indicate the existence of the theory of (pure Grothendieck) motives : there is a semi-simple abelian category of Grothendieck motives M(k) (where k is any ground field) satisfying the properties:

(1) To a smooth projective variety X over k, its motive h(X), an object of M(k), is associated; h(X) is contravariantly functorial in X.

(2) Any Weil cohomology H^{∗} (which is a contravariant functor from the category of smooth projective varieties to the category of graded vector spaces) factors through M(k). The resulting functor (called the realization functor) from M(k) to (graded vector sp.) is exact and faithful. Morally, the vector space H^{∗}(X) together with its functorial behavior is ruled by the motive h(X). The functor h is the universal Weil cohomology taking values in a semi-simple abelian category.

In view of the discovery of mixed Hodge structures by Deligne – the theory of weights for Betti cohomology [De 2], one may naturally suspect the existence of the category of mixed motives – motives with weights – which contains the “motives” of quasi-projective singular varieties. This abelian category is no more semi-simple, which makes it all the more interesting, because extension groups between objects are non-trivial. The significance of such a category was made clear by the conjecture that these extension groups should be related to Chow groups and higher K-groups of varieties, [Be], [De]; see also [Ja].

The purpose of this paper is to construct the category D(k) of mixed motives (over a field k) and study some properties. Such a construction was made possible by incorporating the theory of higher Chow groups due to S. Bloch. We perceive a mixed motive as a diagram built up of smooth projective varieties and correspondences (in the sense of higher Chow groups) between them.

Part of this paper appeared in [Ha 1, Part I]. In [Ha, Part III] we discuss conjectural properties of the category D(k). The paper [Ha 2] treats the materials in §6 more fully. The references at the end are limited; for a more comprehensive bibliography related to mixed motives, the reader may consult [Ja], for example.

We would like to warmly thank S. Bloch for his collaboration on the project. We are also grateful to many other people, particularly A. Corti, W. Fulton, U. Jannsen, M. Levine, R. MacPherson, and J. Murre, for helpful discussions and advice. The author is grateful for the hospitality of the Max-Planck-Institut in Bonn during his stay in 1995-96.

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For other approaches to categorical constructions, see [Le 2] and [Vo]. In our conception, higher correspondences are built into the very definition of motives, while in the other constructions higher Chow groups are revealed later in the theories.

§1 – §3 are preliminary materials. We discuss some results on cycle complexes in §1. A large part is concerned with the notion of distinguished subcomplexes of a cycle complex; on many occasions it is necessary to restrict oneself to a subcomplex on which certain operations are defined. We need to define such distinguished subcomplexes, and show the “moving lemma”: a distinguished subcomplex is quasi-isomorphic to the whole cycle complex. The results are stated in (1.1)-(1.7). The detailed proofs take the rest of the section; the reader may skip them in the first reading.

In §2, we recall the construction of the category of pure motives by Grothendieck. The reason of describing the construction in a way slightly different from the literature – introducing the category of formal symbols – is because it will also appear as an intermediate category to construct D(k).

As a generalization of the notion of double complexes, we introduce C-complexes in §3 in an abstract form. Similar types of objects appear in the definition of mixed motives in §4 and in their cohomology realizations in §5.

We define the category D(k) in §4. Remark both the objects and the morphisms will be quite explicitly given. We also study the structures of D(k).

(4.10) Theorem. The triangulated category D(k) has the following properties:

(1) D(k) has dual, tensor product, inner Hom, the unit object \underline{ℚ}, and the Tate objects \underline{ℚ}(r);

(2) (ch(k) = 0) The contravariant functor h : (Smooth P roj./k) → D(k) can be extended to a functor from the category of quasi-projective varieties; to each quasi-projective X there is associated its (cohomological) motive h(X), an object of D(k);

(3) If X is smooth and projective, one has

Hom_{D(k)}(\underline{ℚ}, h(X)(r)[2r − m]) = K_{m}(X)_{ℚ}^{(r)} .

Here the right hand side is an Adams-graded piece of the K-group of X.

(4) If H^{∗} denotes either of etale or Betti cohomology, there is the corresponding realization functor H^{∗} : D(k) → V ect (Λ); here V ect (Λ) is the category of Λ-vector spaces, and Λ is the coefficient field of the cohomology theory: Λ = ℚ_{ℓ} in the etale case, and Λ = ℚ in the Betti case.

In §5, the cohomology realization functors are constructed for Betti cohomology.

§6 is an exposition of (2) and more. In particular, for a quasi-projective variety over k of characteristic zero, we define its higher Chow cohomology/homology groups with/without compact support.

The conjectural properties of the category D(k) are discussed in Part III, particularly the existence of the expected t-structure. Assuming some “standard conjectures” (slightly more than the conjectures by Grothendieck, Murre and Soulé-Beilinson) it can be proven: There is a unique t-structure on D(k) such that either of Betti or etale cohomology functor ℝΓ : D(k) → D(Λ) (where D(Λ) is the derived category of complexes of Λ-vector spaces) is a t-exact functor with respect to the t-structure and the natural t-structure on D(Λ).

The t-structure would then provide an abelian subcategory of D(k) as its heart; this is the candidate of the abelian category of mixed motives.

§1. Cycle complexes and higher Chow groups.

§2. Grothendieck’s construction

§3. C-complexes.

§4. The category D(k).

§5. Cohomology realizations of motives.

§6. Cohomological motives of quasi-projective varieties.

References.

The following displays categories and functors appearing in this paper.(↪ indicates a fully faithful functor.)

D_{finite}(k)
↗  ↘
Symb_{finite} (k) ↪ CHM(k) ↪ D(k)
↓ ↓ ↓
Symb_{inf} (k) ↪ CHM_{inf}(k) ↪ D_{inf}(k)

There is the “cohomological motive” functor

h : (Smooth P roj/k)^{opp} → D_{finite}(k) .

This factors through Symb_{finite}(k). If ch(k) = 0, h extends to h : (Quasi − P roj/k)^{opp} → D_{finite}(k).

(See Part III) Under certain conjectures, an abelian subcategory (consisting of cohomological degree 0 objects) D(k)^{deg 0} ↪ D(k) exists, and one has a non-commutative square

CHM(k) ↪  D(k)
↓ cano  ↑
M(k)  ↪ D(k)^{deg 0}

where cano is the canonical functor.

§1. Cycle complexes and higher Chow groups.

In this paper k is an arbitrary ground field, and one considers separated schemes of finite type (we will simply say schemes) over k. A variety is a reduced, possibly reducible scheme over k.

The references for cycle complexes are [Bl 1,2,3]. The exposition below will be selfcontained except for the citation of (1.1) and (1.8). For generalities on algebraic cycles and intersection theory, see [Fu]. Among the various versions, we employ the alternating cubical cycle complex [Bl 3], since we need the graded commutativity for the internal product.

Let □^{n} = 𝔸_{k}^{n} with coordinates (x_{1}, ⋯, x_{n}). Faces of □^{n} are intersections of codimension one faces, and the latter are divisors of the form □_{i,a}^{n−1} = {x_{i} = a} where a = 0 or 1. A face of dimension m is canonically isomorphic to □^{m}. Let Σ_{n} be the permutation group of the set {1, ⋯, n} acting on □^{n} by permutations of the coordinates.

Let X be an equi-dimensional variety (or a scheme) over a field k. Let Z^{r}(X, n) be the ℚ-vector space of ℚ-cycles z of codimension r on X × □^{n} such that

(i) each irreducible component of z meets each face X × □^{m} properly, and

(ii) z is alternating with respect to Σ_{n}, namely, for any σ ∈ Σ_{n}, σ(z) = sgn(σ)z. The inclusions of codimension one faces δ_{i,a} : □_{i,a}^{n−1} ↪ □^{n} induce the map

∂ = ∑(−1)^{i+a}δ_{i,a}^{∗} : Z^{r}(X, n) → Z^{r}(X, n − 1)

(namely, ∂ sends alternating cycles to alternating ones) and (Z^{r}(X, ·), ∂) is a homology complex. We call this the cycle complex (of codimension r) of X. By definition the (rational) higher Chow groups are the homology groups of this complex:

CH^{r}(X, n) = H_{n}Z^{r}(X, ·) .

Note CH^{r}(X, 0) = CH^{r}(X), the rational Chow group of X.

Considering ℤ- cycles in X × □^{n} which are “normalized”, in place of ℚ-cycles, integral higher Chow group can be defined. In this paper we only consider rational cycle complexes and rational Chow groups. Rational Chow groups are isomorphic to Adams graded pieces of K-groups of X.

(1.1) Theorem ([Bl 2] [Le 1]). There is a canonical isomorphism

CH^{r}(X, n) ≅ K′_{n}(X)_{ℚ}^{(r)}

where K′_{n}(X) is the K-group of the exact category of coherent sheaves on X, K′_{n}(X)_{ℚ} = K′_{n}(X) ⊗ ℚ, and K′_{n}(X)_{ℚ}^{(r)} is the r-th Adams eigenspace.

For the rest of the section, all varieties are equi-dimensional. A proper map of varieties f : X → Y induces a map of cycle complexes f_{∗} : Z^{r}(X, ·) → Z^{r+dim Y −dim X}(Y, ·). If f : X → Y is a flat map, there is induced a map f^{∗} : Z^{r}(Y, ·) → Z^{r}(X, ·). If Y is smooth, there is a quasi-isomorphic subcomplex Z^{r}(Y, ·)′ ⊂ Z^{r}(Y, ·) on which f^{∗} : Z^{r}(Y, ·)′ → Z^{r}(X, ·) is defined.

For more general results, valid only for smooth varieties, consider the following operations. Let X, Y, Z be smooth projective varieties. For cycles z and w on a smooth variety, we will denote cycle theoretic intersection by z ⊙ w or z · w.

The map induced by a correspondence. For an element f ∈ Z^{s}(X × Y, ℓ), the partially defined map (of graded vector spaces)

f_{∗} : Z^{r}(X, n) − − → Z^{r+s}(Y, n + ℓ), f_{∗}(z) = p_{Y} _{∗}[(f × □^{n}) · (z × Y × □^{ℓ})] .

Here (f × □^{n}) · (z × Y × □^{ℓ}) denotes the alternation (with respect to the action of Σ_{n+ℓ}) of the cycle theoretic intersection (f ×□^{n})⊙(z ×Y ×□^{ℓ}), and p_{Y} : X ×Y ×□^{n+ℓ} → Y ×□^{n+ℓ} is the projection.

One has the formula

(∂f)_{∗} = ∂∘f_{∗} − (−1)^{ℓ−1}f_{∗}∘∂

whenever all the terms are defined.

Transpose. The involution

Z^{r}(X × Y, ·) →^{∼} Z^{r}(Y × X, ·), z ↦ ^{t}z := s_{∗}z

where s : X × Y → Y × X is the isomorphism which switches the factors.

External product. The map

Z^{r}(X, ·) ⊗ Z^{s}(Y, ·) → Z^{r+s}(X × Y, ·)

which sends z⊗w to z×w. (More precisely, if z ∈ Z^{r}(X, n) and w ∈ Z^{s}(Y, m) z×w is a cycle on X × □^{n} × Y × □^{m}. The isomorphism which switches the factors s : X × □^{n} × Y × □^{m} →^{∼} X × Y × □^{n} × □^{m} gives the cycle s_{∗}(z × w) on X × Y × □^{n+m}; take then its alternation with respect to Σ_{n+m}. When there is no confusion, denote this also by z × w by abuse of notation.) This is associative and a map of complexes, namely satisfies Leibniz formula

∂(z × w) = (∂z) × w + (−1)^{deg z} z × (∂w) .

The transpose is graded anti-commutative:

t(z × w) = (−1)^{deg z deg w} ^{t}w ×^{t}z .

Internal product. The partially defined map

Z^{r}(X, n) ⊗ Z^{s}(X, m) − − → Z^{r+s}(X, n + m)

which sends z ⊗ w to

z · w := (z × □^{m}) · (w × □^{n}) := the alternation of (z × □^{m}) ⊙ (w × □^{n}) .

This is associative, graded commutative and satisfies Leibniz formula. The graded commutativity reads: w · z = (−1)^{deg w deg z}z · w.

Composition of correspondences. The partially defined map

Z^{r}(X × Y, ·) ⊗ Z^{s}(Y × Z, ·) − − → Z^{r+s−dim Y} (X × Z, ·)

which sends u ⊗ v ∈ Z^{r}(X × Y, n) ⊗ Z^{s}(Y × Z, m) to

v∘u := the alternation of p_{XZ ∗}[(X × v × □^{n}) ⊙ (u × Z × □^{m})] .

(Here again, u × Z × □^{m}, for example, is a cycle on X × Y × □^{n} × Z × □^{m} and one regards it as a cycle on X × Y × Z × □^{n+m}. For the rest of the paper, we will use similar abuse of notation without further explanation.) This is associative and satisfies Leibniz formula. One also has

t(u∘v) = (−1)^{deg u deg v} ^{t}v∘^{t}u .

Relations among the above operations. If f : X → Y is a map of smooth projective varieties, and Γ_{f} ⊂ Y × X its graph, one has

(Γ_{f})_{∗} = f^{∗} and (^{t} Γ_{f})_{∗} = f_{∗} .

If f ∈ Z(X × Y, ℓ) and g ∈ Z(Y × Z, m), then

g_{∗}f_{∗} = (g∘f)_{∗} .

If u ∈ Z(X × Y, ℓ) and v ∈ Z(Y × Z, m), then

v∘u = (Δ_{X} × v)_{∗}u

where Δ_{X} × v ∈ Z((X × Y) × (X × Z), m). Similarly, v∘u = (^{t}u × Δ_{Z})_{∗}v.

Remark. (1) The notation z×w is for both the usual external product and its alternation; in each occasion it will be clear which we mean.

(2) Some of the above operations are defined for smooth quasi-projective varieties. For example, for the map f_{∗} be defined, one only needs X smooth projective and Y smooth quasi-projective; external and internal products are defined for smooth quasi-projective X. For the rest of the section also, one may often work under weaker assumptions on varieties. Yet we will not be concerned with that, and assume varieties X, Y, etc. are smooth projective (unless otherwise stated); it will suffice for our purposes.

Two subvarieties V, W on a equi-dimensional variety X intersect properly if dim(V ∩ W) ≤ dim V + dim W − dim X. We define that two cycles α, β on X intersect properly if each component of α and each component of β intersect properly. We will denote proper intersection by α P ⌢ β.

For a set I = {(a, i)} ⊂ {0, 1} × [m], such that for each i, there is at most one a such that (a, i) ∈ I, there is the corresponding face □_{I}^{m−|I|} = {x_{i} = a for (a, i) ∈ I} ⊂ □^{m} and the boundary map ∂_{I} : Z(X, m) → Z(X, m − |I|).

Let X be a smooth projective variety. Let Y be a smooth projective variety and W = {W_{i}} a finite set where W_{i} is an irreducible closed set of X ×Y ×□^{ℓ_{i}} meeting faces properly. Let Z_{W}^{r} (X, ·) be the subcomplex of Z^{r}(X, ·) generated by irreducible cycles z in Z^{r}(X, n) satisfying

z × Y × □^{ℓ_{i}} P ⌢ W_{i} × □^{n} in X × Y × □^{n+ℓ_{i}} and
(z × Y × □^{ℓ_{i}}) ⊙ (W_{i} × □^{n}) P ⌢ any face of X × Y × □^{n+ℓ_{i}}

where (z × Y × □^{ℓ_{i}}) ⊙ (W_{i} × □^{n}) is the cycle theoretic intersection, not its alternation. When there is no confusion, we will simply put “(z × Y × □^{ℓ_{i}}) · (W_{i} × □^{n}) is defined and ∈ Z(X × Y, n + ℓ_{i}).” The condition is equivalent to:

For any faces □_{J} ⊂ □^{ℓ_{i}} and □^{n′} ⊂ □^{n},
z × Y × □^{ℓ_{i}} P ⌢ ∂_{J}W_{i} × □^{n′} in X × Y × □^{n+ℓ_{i}} .

The equivalence follows from the following lemma (readily verified and often to be used).

(1.2) Lemma. Let A, B and C be effective algebraic cycles on a quasi-projective variety X such that A P ⌢ B, B P ⌢ C, and A P ⌢ C. Then

A · C P ⌢ B ⇔ A · B P ⌢ C .

A subcomplex of the form Z_{W}^{r} (X, ·) is called a distinguished subcomplex. We will use the notation Z^{r}(X, ·)′ if it is not necessary to specify W. Note that the intersection of a finite collection of distinguished subcomplexes is again distinguished. The proof of the following is in (1.12).

(1.3) Proposition. For a smooth projective variety X the inclusion of a distinguished subcomplex Z^{r}(X, ·)′ → Z^{r}(X, ·) is a quasi-isomorphism.

(1.4) Proposition. (1) Let f ∈ Z^{s}(X × Y, ℓ). There is a distinguished subcomplex Z^{r}(X, ·)′ ⊂ Z^{r}(X, ·) such that the map of graded groups

f_{∗} : Z^{r}(X, n)′ → Z^{r+s}(Y, n + ℓ) , f_{∗}(z) = p_{Y} _{∗}[(f × □^{n}) · (z × Y × □^{ℓ})]

is defined. (Recall (f × □^{n}) · (z × Y × □^{ℓ}) is the alternation of cycle theoretic intersection.) (2) Let w ∈ Z^{s}(X, ℓ). There is a distinguished subcomplex Z^{r}(X, ·)′ such that the map

w· : Z^{r}(X, n)′ → Z^{r+s}(X, n + ℓ) , z ↦ w · z := (w × □^{n}) · (z × □^{ℓ})

is defined.

(3) Let v ∈ Z^{s}(Y × Z, ℓ). There is a distinguished subcomplex Z^{r} (X × Y, ·)′ such that the map

v∘ : Z^{r}(X × Y, n)′ → Z^{r+s−dim Y} (X × Z, n + ℓ) , v∘u := p_{XZ}_{∗}[(X × v × □^{n}) · (u × Z × □^{ℓ})]

is defined.

One has a similar statement for w ∈ Z^{s}(Z × X, ℓ) and the map ∘w.

Proof. (1) Clearly, we can take Z^{r}(X, ·)′ = Z_{{f}}^{r}(X, ·).

(2) Let δ : X → X × X be the diagonal map. Then take Z^{r}(X, ·)′ to be the distinguished subcomplex with respect to the cycle δ_{∗}(w) on X × X × □^{ℓ}.

(3) Take Z^{r}(X ×Y, ·)′ to be the distinguished subcomplex with respect to the cycle X ×v on X × Y × Z × □^{ℓ}.

The following amplification holds. It will be used in §4. Let |f| be the support of a cycle f. Suppose two cycles f ∈ Z(X × Y, ℓ) and g ∈ Z(Y × Z, m) satisfy: |f| × Z × □^{m} P ⌢ X × |g| × □^{ℓ} in X × Y × Z × □^{ℓ+m}, and their cycle theoretic intersection meets properly the faces. We then say the composition g∘f is defined and ∈ Z(X × Z, ℓ + m). Let g∘f = (p_{XZ})_{∗}[(f × Z × □^{m}) · (X × g × □^{ℓ})] ∈ Z(X × Z, ℓ + m).

(1.5) Proposition. (1) Let f ∈ Z(X × Y, ℓ), g ∈ Z(Y × Z, m) such that g∘f is defined and ∈ Z(X × Z, ℓ + m). Then there are distinguished subcomplexes Z(X, ·)′ and Z(Y, ·)′ such that the three maps

f_{∗} : Z(X, ·)′ → Z(Y, · + ℓ)′, g_{∗} : Z(Y, ·)′ → Z(Z, · + m), and
(g∘f)_{∗} : Z(X, ·)′ → Z(Z, · + ℓ + m)

are all defined and (g∘f)_{∗} = g_{∗}∘f_{∗}.

(2) Let v ∈ Z(Y ×Z, ℓ), w ∈ Z(Z ×W, m) such that w∘v is defined and ∈ Z(Y ×W, ℓ+m). Then there are distinguished subcomplexes Z(X × Y, ·)′ and Z(Y × Z, ·)′ such that the three maps

v∘ : Z(X × Y, ·)′ → Z(X × Z, · + ℓ)′, w∘ : Z(X × Z, ·)′ → Z(X × W, · + m), and

(w∘v)∘ : Z(X × Y, ·)′ → Z(X × W, · + ℓ + m)

are all defined and (w∘)(v∘) = (w∘v)∘. (There is a similar statement for compositions “from the right”.)

Remark. These can be generalized to the case where a finite sequence of composable correspondences are given. For example let W be another smooth projective variety and f ∈ Z(X × Y, ℓ), g ∈ Z(Y × Z, m) and h ∈ Z(Z × W, n). We assume: g∘f is defined and ∈ Z(X × Z, ℓ + m), h∘g is defined and ∈ Z(Y × W, m + n), and h∘g∘f is defined and ∈ Z(X × W, ℓ + m + n). (The last condition, by definition, is the following.

(|f| × Z × W × □^{m+n}) ⊙ (X × |g| × W × □^{ℓ+n}) P ⌢ X × Y × |h| × □^{ℓ+m}

and the intersection meeting properly the faces of X × Y × Z × W × □^{ℓ+m+n}. ) Then there are distinguished subcomplexes Z(X, ·)′, Z(Y, ·)′ and Z(Z, ·)′ such that the maps

Z(X, ·)′ ⟶^{f_{∗}} Z(Y, · + ℓ)′ ⟶^{g_{∗}} Z(Z, · + ℓ + m)′ ⟶^{h_{∗}} Z(W, · + ℓ + m + n)

are all defined, and

g_{∗}f_{∗} = (g∘f)_{∗}, h_{∗}g_{∗} = (h∘g)_{∗}, and h_{∗}g_{∗}f_{∗} = (h∘g∘f)_{∗} .

Similarly for (2). (The proof below is for the case of two correspondences; the general case is similar.)

Proof. (1) Working with irreducible components separately, we may assume f and g are irreducible. Since g∘f is defined,

f × Z × □^{m} P ⌢ X × g × □^{ℓ} in X × Y × Z × □^{ℓ+m} .

Take the distinguished subcomplex Z(X, ·)′ ⊂ Z(X, ·) with respect to the set

{f × Z, X × g, (f × Z × □^{m}) ⊙ (X × g × □^{ℓ})}

of cycles on X ×Y ×Z ×□^{?} (the third one is the intersection theoretic one, not its alteration) and the distinguished subcomplex Z(Y, ·)′ with respect to {g} on Y × Z × □^{m}. We claim these satisfy the required conditions.

For z ∈ Z(X, n)′, assumed irreducible, we need to show

(*) p_{Y ∗}[(f × □^{n}) ⊙ (z × Y × □^{ℓ})] × Z × □^{m} P ⌢ g × □^{n+ℓ} in Y × Z × □^{n+m+ℓ}

and more generally

(**) p_{Y ∗}[(f ×□^{n})⊙(z×Y ×□^{ℓ})]×Z ×□^{m} P ⌢ ∂_{K}g×□_{J} ×□^{n′} in Y ×Z ×□^{n+m+ℓ} .

for faces □_{J} ⊂ □^{ℓ}, □_{K} ⊂ □^{m} and □^{n′} ⊂ □^{n}.

From

(z × Y × Z × □^{ℓ+m}) P ⌢ (f × Z × □^{n+m}) · (X × g × □^{n+ℓ})

using Lemma (1.2) we have

(z × Y × Z × □^{ℓ+m}) ⊙ (f × Z × □^{n+m}) P ⌢ X × g × □^{n+ℓ} .

By the lemma below (easily verified) applied to the projection p : X × Y × Z × □^{n+m+ℓ} → Y × Z × □^{n+m+ℓ} we obtain (*). The proof of (**) is similar. (2) is proved in a similar manner.

Lemma. Let W be a quasi-projective variety, X an equi-dimensional projective variety and p : X × W → W the projection. For algebraic cycles u on X × W and w on W,

u P ⌢ X × w ⇒ p_{∗}u P ⌢ w .

We now consider the tensor product of cycle complexes

Z^{r}(X, ·) ⊗ Z^{s}(Y, ·)

where X, Y are smooth projective. Let W = {W_{i}} be such that for a smooth projective Z, W_{i} are irreducible closed subsets of X × Y × Z × □^{ℓ_{i}} meeting faces properly. Define the subcomplex

[Z^{r}(X, ·) ⊗ Z^{s}(Y, ·)]_{W}

as the one generated by z ⊗ w ∈ Z^{r}(X, n) ⊗ Z^{s}(Y, m), with z, w irreducible, satisfying the three conditions: for any faces □_{J} ⊂ □^{ℓ_{i}}, □^{n′} ⊂ □^{n} and □^{m′} ⊂ □^{m},

z × Y × Z × □^{ℓ_{i}} P ⌢ ∂_{J}W_{i} × □^{n′} in X × Z × □^{n+ℓ_{i}} ,
X × w × Z × □^{ℓ_{i}} P ⌢ ∂_{J}W_{i} × □^{m′} in Y × Z × □^{m+ℓ_{i}} ,
z × w × Z × □^{ℓ_{i}} P ⌢ ∂_{J}W_{i} × □^{n′+m′} in X × Y × Z × □^{n+m+ℓ_{i}} .

The first two conditions say z ∈ Z_{W} (X) and w ∈ Z_{W} (Y).

A subcomplex of this form is called a distinguished subcomplex. The intersection of a finite number of distinguished subcomplexes is distinguished. It is easy to verify that if Z^{r}(X, ·)′ ⊂ Z^{r}(X, ·) and Z^{s}(Y, ·)′ ⊂ Z^{s}(Y, ·) are distinguished subcomplexes, then Z^{r}(X, ·)′ ⊗ Z^{s}(Y, ·)′ is a distinguished subcomplex.

We again postpone the proof of the following until (1.13).

(1.6) Proposition. The inclusion of a distinguished complex into Z^{r}(X, ·) ⊗ Z^{s}(Y, ·) is a quasi-isomorphism.

(1.7) Proposition. (1) For any distinguished subcomplex Z^{r+s}(X×Y, ·)′ ⊂ Z^{r+s}(X×Y, ·), there is a distinguished subcomplex [Z^{r}(X, ·) ⊗ Z^{s}(Y, ·)]′ such that the external product

[Z^{r}(X, ·) ⊗ Z^{s}(Y, ·)]′ → Z^{r+s}(X × Y, ·)′ , z ⊗ w ↦ z × w

is defined.

(2) For a distinguished subcomplex Z^{r+s}(X, ·)′, there is a distinguished subcomplex [Z^{r}(X, ·) ⊗ Z^{s}(X, ·)]′ such that the internal product

[Z^{r}(X, ·) ⊗ Z^{s}(X, ·)]′ → Z^{r+s}(X, ·)′ , z ⊗ w ↦ z · w

is defined.

(3) For a distinguished subcomplex Z^{r+s}(X × Z, ·)′, there is a distinguished subcomplex [Z^{r}(X × Y, ·) ⊗ Z^{s}(Y × Z, ·)]′ such that the composition

[Z^{r}(X × Y, ·) ⊗ Z^{s}(Y × Z, ·)]′ → Z^{r+s}(X × Z, ·)′ , u ⊗ v ↦ v∘u

is defined.

Proof. (1) There is a smooth variety Z and cycles W_{i} on X × Y × Z × □^{ℓ_{i}} such that Z^{r+s}(X × Y, ·)′ = Z_{W}^{r+s}(X × Y, ·). Then one may take the distinguished subcomplex [Z^{r}(X, ·) ⊗ Z^{s}(Y, ·)]_{W}.

(2) Let W_{i} are on X × Y × □^{ℓ_{i}} such that Z^{r+s}(X, ·)′ = Z_{W}^{r+s}(X, ·). Then take as [Z^{r}(X, ·) ⊗ Z^{s}(X, ·)]′ the distinguished subcomplex with respect to the collection {δ_{∗}W_{i}} on X × X × Y × □^{ℓ_{i}} (δ : X × Y → X × X × Y is the diagonal).

(3) Similar to (1), (2) and left to the reader.

The rest of §1 is concerned with the proof of (1.3) and (1.6). It will be an adaptation of [Bl 1]. For example, (1.3) is proven there in the case Y = pt and ℓ_{i} = 0. We first show (1.3) in the special case X = ℙ^{N}. The following lemma is from loc. cit.

(1.8) Lemma. Let G be a connected algebraic group acting on an equi-dimensional variety X. Let A, B be equi-dimensional closed subsets of X. Assume

(1.8.1) the fibers of the map σ : G × A → X have the same dimension, and
\overline{σ(G × A)} P ⌢ B in X .

Then there is an non-empty open subset U ⊂ G such that for any g ∈ U(k̄), g(A) P ⌢ B in X.

In fact, in [Bl 1, (1.1)], it is assumed that the map G × A → X is dominant. But the same proof shows the above. The result below is [Bl 1, (1.2)], except again the hypothesis is weaker.

(1.9) Proposition. Under the same hypothesis as in (1.8), assume further given a field extension K/k, and a K-morphism ψ : X_{K} → G_{K}. Let ∅ ≠ V ⊂ X be an open set such that for every point x ∈ V_{K} we have

(1.9.1)  tr deg_{k} k(ψ(x), x) ≥ dim G .

Define ϕ : X_{K} → X_{K} by ϕ(x) = ψ(x) · x, and assume ϕ is an isomorphism. Then the intersection ϕ(A ∩ V) ∩ B is proper.

Remark. In one of the applications, we have a variety X with G acting on it, and given a map ψ : 𝔸_{K}^{1} → G_{K} such that for a ∈ 𝔸_{K}^{1}, a ≠ 0, tr deg_{k} ψ(a) ≥ dim G − 1, and if a is algebraic over k, ψ(a) is k-generic in G. Then we take the ambient variety X × 𝔸^{1} with G acting on the first factor, the map ψ∘ pr_{2} : X × 𝔸^{1} → 𝔸^{1} → G, and the open set V = X × (𝔸^{1} − {0}). Then the assumption on the transcendence degree (1.9.1) is satisfied.

(1.10) Proposition. Let W = {W_{i}}, with W_{i} cycles on ℙ^{N} × Y × □^{ℓ_{i}} meeting faces properly. Then the inclusion Z_{W}^{r} (ℙ^{N}, ·) ⊂ Z^{r}(ℙ^{N}, ·) is a quasi-isomorphism.

Proof. Let X = ℙ_{k}^{N}, and K be a purely transcendental function field over k. Let π : X_{K} → X be the canonical morphism, and

π^{∗} : Z^{r}(X, ·) → Z^{r}(X_{K}, ·)

be the pull-back by π. Since it induces π^{∗} : Z_{W}^{r} (X, ·) → Z_{W_{K}}^{r} (X_{K}, ·), there is the induced map of complexes

π^{∗} : Z^{r}(X, ·)/Z_{W}^{r} (X, ·) → Z^{r}(X_{K}, ·)/Z_{W_{K}}^{r} (X_{K}, ·) .

If we show π^{∗} is injective and induces zero on homology, the acyclicity of Z^{r}(X, ·)/Z_{W}^{r} (X, ·) will follow.

The injectivity is verified as follows. First assume k is infinite and K = k(X_{1}, ⋯ X_{n}). If z ∈ Z^{r}(X, ·) satisfies

π^{∗}z = z_{1} + ∂w, z_{1} ∈ Z_{W}^{r} (X_{K}, ·), and w ∈ Z^{r}(X_{K}, · + 1)

then for an non-empty open set U ⊂ 𝔸_{k}^{n} and x ∈ U(k), the specializations satisfy z_{1}(x) ∈ Z_{W}^{r} (X, ·), w(x) ∈ Z^{r}(X, · + 1) and

z = z_{1}(x) + ∂w(x) .

In general, note if k′/k is a finite field extension, one has maps

Z^{r}(X_{k}, ·) ⟶^{π^{∗}} Z^{r}(X_{k′}, ·) ⟶^{π_{∗}} Z^{r}(X_{k}, ·)

and π_{∗}π^{∗} = [k′ : k]. Consider an infinite tower of field extensions of p-power degrees (for two different primes p) and apply the above argument.

We have to show π^{∗} induces zero on homology. Take a purely transcendental function field K over k, and a map ψ : 𝔸_{K}^{1} → G_{K} so that ψ(0) is the identity of G(K), and ψ satisfies the condition of (1.9), Remark. Let pr_{1} : X × 𝔸^{1} → X be the projection and let

ϕ : X × 𝔸^{1} → X × 𝔸^{1}

be the isomorphism over K given by (x, a) ↦ (ψ(a).x, a). Let G act on X ×𝔸^{1} by g.(x, a) = (g.x, a). Set

ρ^{∗} = ϕ^{∗} pr_{1}^{∗} π^{∗} : Z^{r}(X_{k}, ·) → Z^{r}(X_{K}, · + 1) .

Claim. (1) ψ(1)^{∗} sends Z^{r}(X_{k}, ·) into Z_{W}^{r} (X_{K}, ·).

(2) ρ^{∗} sends Z_{W}^{r} (X_{k}, ·) into Z_{W}^{r} (X_{K}, · + 1).

Note that the claim implies that ρ^{∗} induces a map

Z^{r}(X_{k}, ·)/Z_{W}^{r} (X_{k}, ·) → Z^{r}(X_{K}, · + 1)/Z_{W}^{r} (X_{K}, · + 1)

which provides a homotopy to zero

π^{∗} ≃ ψ(1)^{∗} = 0 : Z^{r}(X_{k}, ·)/Z_{W}^{r} (X_{k}, ·) → Z^{r}(X_{K}, ·)/Z_{W}^{r} (X_{K}, ·) .

For Claim (1), we take z ∈ Z(X_{k}, n), assumed irreducible, and must show for any faces □_{J} ⊂ □^{ℓ} and □^{n′} ⊂ □^{n},

ψ(1)^{∗}(z × Y × □^{ℓ}) P ⌢ ∂_{J}W × □^{n′} in X × Y × □^{n} × □^{ℓ} .

We apply (1.9) to the ambient variety X × Y × □^{n+ℓ}, subvarieties A = z × Y × □^{ℓ}, and B = ∂_{J}W × □^{n′}. For the map ψ, we take the constant map ψ(1) : X × Y × □^{n+ℓ} → G.

We must verify the condition (1.8.1). The fibers of σ : G × A → X × Y × □^{n+ℓ} are isomorphic to the stabilizers of points of A, and they all have the same dimension. Also, we have that

σ(G × A) = X × pr(z) × Y × □^{ℓ}

(pr : X × □^{n} → □^{n} is the projection). The assertion σ(G × A) P ⌢ B is equivalent to pr(z) P ⌢ □^{n′} in □^{n}, which easily follows from z P ⌢ X × □^{n′} in X × □^{n}.

For (2), take z ∈ Z_{W} (X_{k}, n), assumed irreducible, and faces F ⊂ 𝔸^{1} × □^{n}, □_{J} ⊂ □^{ℓ}. We will show

(*)  ϕ^{∗}(z × Y × 𝔸^{1} × □^{ℓ}) P ⌢ ∂_{J}W × F in X × Y × 𝔸^{1} × □^{n+ℓ} .

Here and below, we write W and ℓ for W_{i} and ℓ_{i}. ϕ is being used for the automorphism

ϕ × id_{Y ×□^{n+ℓ}} : X × Y × 𝔸^{1} × □^{n+ℓ} → X × Y × 𝔸^{1} × □^{n+ℓ} .

If F = {0}×□^{n′} with □^{n′} ⊂ □^{n}, this is clear since z ∈ Z_{W} (X, n). Suppose F = {1}×□^{n′}. The condition (*) is equivalent to

ψ(1)^{∗}(z × Y × □^{ℓ}) P ⌢ ∂_{J}W × □^{n′} in X × Y × □^{n} × □^{ℓ} ;

this is Claim (1).

We now consider the least obvious case F = 𝔸^{1} × □^{n′} and show

ϕ^{∗}(z × Y × 𝔸^{1} × □^{ℓ}) P ⌢ ∂_{J}W × 𝔸^{1} × □^{n′} in X × Y × 𝔸^{1} × □^{n+ℓ} .

We apply Proposition (1.9), Remark, to the ambient space X × Y × □^{n+ℓ} × 𝔸^{1}, the subvarieties

A = z × Y × □^{ℓ} × 𝔸^{1} and B = ∂_{J}W × □^{n′} × 𝔸^{1} ,

and the map ψ∘pr_{𝔸^{1}} : X × Y × □^{n+ℓ} × 𝔸^{1} → 𝔸^{1} → G. The hypothesis (1.8.1) is satisfied: σ(G×A) = X ×pr(z)×Y ×□^{ℓ} ×𝔸^{1} meets ∂_{J}W ×□^{n′} ×𝔸^{1} properly in X ×Y ×□^{n+ℓ} ×𝔸^{1}, since pr(z) P ⌢ □^{n′} in □^{n}.

Let X be a smooth projective variety. Define, for a cycle z = ∑ a_{i} z_{i} and a subvariety Y ⊂ X,

e(z) = Max_{i}(dim(Y ∩ z_{i}) − dim z_{i} − dim Y + dim X) .

Note e(z) ≥ 0 and e(z) = 0 iff z P ⌢ Y. We recall the moving lemma in [Ro] to the effect that “coning operation” on cycles improves the intersection properties.

Take a closed embedding j : X ↪ ℙ^{N} with N > dim X. Let F be the function field of the Grassmannian variety Grass(N, N − dim X − 1), and L ⊂ ℙ_{F}^{N} be the restriction of the universal family of planes.

For a cycle z on X, let C_{L}(z) be its cone with center L; it is a cycle on ℙ^{N} defined over F meeting X properly. Also let σ_{L}(z) = z − C_{L}(z) ∩ X; it is a cycle on X defined over F.

(1.11) Lemma. In this situation, one has

e(σ_{L}(z) ) ≤ Max (e(z) − 1, 0) .

Corollary. There is a sequence of finitely generated purely transcendental extensions k = K_{0} ⊂ K_{1} ⊂ ⋯ ⊂ K_{r} = K and subspaces L_{i} ⊂ ℙ_{K_{i}}^{N} such that: for any k-cycle z on X and any subvariety Y ⊂ X, one has

σ_{L_{r}} ⋯ σ_{L_{1}}(z) P ⌢ Y in X.

(1.12) Proof of Proposition (1.3). Keeping the above notation let

X_{K_{r}} ⟶^{π_{r}} X_{K_{r−1}} → ⋯ → X_{1} ⟶^{π_{1}} X

be the projections. Consider the maps

π_{i}^{∗} : Z^{r}(X_{K_{i−1}}, ·)/Z_{W}^{r} (X_{K_{i−1}}, ·) → Z^{r}(X_{K_{i}}, ·)/Z_{W}^{r} (X_{K_{i}}, ·) .

By a specialization argument as in (1.10), this map is injective on homology. We will show that π_{r}^{∗}∘ ⋯ ∘π_{1}^{∗} induces zero on homology, so Z^{r}(X, ·)/Z_{W}^{r} (X, ·) is acyclic.

If z ∈ Z^{r}(X, ·) (resp. ∈ Z_{W}^{r} (X, ·)) and irreducible, then C_{L_{1}}(z) ∈ Z_{{X}}^{r}(ℙ^{N}, ·) (resp. ∈ Z_{{W,X}}^{r}(ℙ^{N}, ·)) so one has the first map in the following:

Z^{r}(X, ·)/Z_{W}^{r} (X, ·) →^{C_{L_{1}}} Z_{{X}}^{r}(ℙ_{K_{1}}^{N}, ·)/Z_{{X,W }}^{r}(ℙ_{K_{1}}^{N}, ·) →^{j^{∗}} Z^{r}(X_{K_{1}}, ·)/Z_{W}^{r} (X_{K_{1}}, ·) .

The map j^{∗} is the restriction by j. By the previous proposition, the complex in the middle is acyclic, so the composition j^{∗}C_{L_{1}} induces zero on homology. The same holds for j^{∗}C_{L_{i}}. So the maps

π^{∗} = π_{r}^{∗} ⋯ π_{1}^{∗} and σ_{L_{r}} ⋯ σ_{L_{1}} : Z^{r}(X, ·)/Z_{W}^{r} (X, ·) → Z^{r}(X_{K}, ·)/Z_{W}^{r} (X_{K}, ·)

induce the same map on homology. But the second map is zero, since σ_{L_{r}} ⋯ σ_{L_{1}} takes z ∈ Z(X, n) into Z_{W} (X_{K}, n) by (1.11), Corollary applied to the fibers of z → □^{n} at generic points of faces of □^{n}. We are done.

(1.13) Proof of Proposition (1.6). As for (1.3), we first consider the special case X = ℙ_{k}^{N} and Y = ℙ_{k}^{N′}. We will take finitely generated purely transcendental extensions K/k and E/K, and consider the map of complexes

π_{X}^{∗} ⊗ π_{Y}^{∗} : Z^{r}(X_{k}, ·) ⊗ Z^{s}(Y_{k}, ·) → Z^{r}(X_{K}, ·) ⊗ Z^{s}(Y_{E}, ·)

where π_{X} : X_{K} → X_{k} and π_{Y} : Y_{E} → Y_{k}. Denote the first complex by C(X_{k}, Y_{k}), and its distinguished subcomplex with respect to W by C(X_{k}, Y_{k})_{W}. Similarly, the second complex is denoted by C(X_{K}, Y_{E}) and let C(X_{K}, Y_{E})_{W} be the subcomplex consisting of elements whose images in C(X_{E}, Y_{E}) are in C(X_{E}, Y_{E})_{W}. Since π_{X}^{∗} ⊗ π_{Y}^{∗} sends C(X_{k}, Y_{k})_{W} into C(X_{K}, Y_{E})_{W}, the above map passes to a map of quotient complexes

π_{X}^{∗} ⊗ π_{Y}^{∗} : C(X_{k}, Y_{k})/C(X_{k}, Y_{k})_{W} → C(X_{K}, Y_{E})/C(X_{K}, Y_{E})_{W} .

We will show this map is injective and zero on homology, whence the acyclicity of C(X_{k}, Y_{k})/C(X_{k}, Y_{k})_{W}. The injectivity is again verified by a specialization argument.

Let G = SL_{N+1}, G′ = SL_{N′+1}. Take an extension K/k and ψ_{1} : 𝔸_{K}^{1} → G_{K} as before. Take a further extension E/K and ψ_{2} : 𝔸_{E}^{1} → G′_{E} with similar properties. Let g_{1} = ψ_{1}(1), g_{2} = ψ_{2}(1). Define

ρ_{1}^{∗} : Z^{r}(X_{k}, ·) → Z^{r}(X_{K}, · + 1) and ρ_{2}^{∗} : Z^{s}(Y_{k}, ·) → Z^{s}(Y_{E}, · + 1)

as in (1.10).

Consider the map of degree zero

g_{1}^{∗} ⊗ g_{2}^{∗} : C(X_{k}, Y_{k}) → C(X_{K}, Y_{E})

and maps of degree one

ρ_{1}^{∗} ⊗ id , g_{1}^{∗} ⊗ ρ_{2}^{∗} : C(X_{k}, Y_{k}) → C(X_{K}, Y_{E}) .

Claim. (1) g_{1}^{∗} ⊗ g_{2}^{∗} sends C(X_{k}, Y_{k}) into C(X_{K}, Y_{E})_{W}.

(2) ρ_{1}^{∗} ⊗ id and g_{1}^{∗} ⊗ ρ_{2}^{∗} sends C(X_{k}, Y_{k})_{W} into C(X_{K}, Y_{E})_{W}.

From the claim it follows that the map (of degree one)

ρ_{1}^{∗} ⊗ id + g_{1}^{∗} ⊗ ρ_{2}^{∗} : C(X_{k}, Y_{k})/C(X_{k}, Y_{k})_{W} → C(X_{K}, Y_{E})/C(X_{K}, Y_{E})_{W}

is defined and gives homotopy

π_{X}^{∗} ⊗ π_{Y}^{∗} ≃ g_{1}^{∗} ⊗ g_{2}^{∗} = 0 : C(X_{k}, Y_{k})/C(X_{k}, Y_{k})_{W} → C(X_{K}, Y_{E})/C(X_{K}, Y_{E})_{W} .

For (1), we take z ⊗ w ∈ C(X_{k}, Y_{k}), z and w irreducible, and must show

g_{1}^{∗}z × g_{2}^{∗}w × Z × □^{ℓ} P ⌢ ∂_{J}W × □^{n′+m′} in X × Y × Z × □^{n+m+ℓ} .

Apply (1.9) to the ambient variety X ×Y ×Z ×□^{n+m+ℓ}, the subvarieties A = z×w×Z ×□^{ℓ}, B = ∂_{J}W ×□^{n′+m′}, and the constant map (g_{1}, g_{2}) : X ×Y ×Z ×□^{n+m+ℓ} → G×G′. (1.8.1) is verified as follows. We have

σ(G × G′ × A) = X × pr(z) × Y × pr(w) × Z × □^{ℓ}

where pr(z) ⊂ □^{n} and pr(w) ⊂ □^{m} are projections of z and w. Thus

σ(G × G′ × A) P ⌢ ∂_{J}W × □^{n′+m′}
⇔ pr(z) P ⌢ □^{n′} in □^{n} and pr(w) P ⌢ □^{m′} in □^{m}
⇐ z P ⌢ X × □^{n′} in X × □^{n} and w P ⌢ Y × □^{m′} in Y × □^{m} .

For (2), take z ⊗ w ∈ C(X_{k}, Y_{k}), z and w irreducible, and we show for any face F ⊂ 𝔸^{1} × □^{n+m+ℓ}

(1.13.1)  ρ_{1}^{∗}z × w × Z × □^{ℓ} P ⌢ ∂_{J}W × F in X × Y × Z × 𝔸^{1} × □^{n+m+ℓ} .

If F = {0} × □^{n′+m′}, (1.13.1) is equivalent to

z × w × Z × □^{ℓ} P ⌢ ∂_{J}W × □^{n′+m′} in X × Y × Z × □^{n+m+ℓ} ,

which holds as z ∈ Z_{W} (X).

If F = {1} × □^{n′+m′}, (1.13.1) is equivalent to

(1.13.2)  g_{1}^{∗}z × w × Z × □^{ℓ} P ⌢ ∂_{J}W × □^{n′+m′} in X × Y × Z × □^{n+m+ℓ} .

We have X × w × Z × □^{n+ℓ} P ⌢ ∂_{J}W × □^{n′+m′} by assumption. Also

g_{1}^{∗}z × Y × Z × □^{m+ℓ} P ⌢ ∂_{J}W × □^{n′+m′}

and

g_{1}^{∗}z × Y × Z × □^{m+ℓ} P ⌢ (X × w × Z × □^{n+ℓ}) · (∂_{J}W × □^{n′+m′})

as can be shown by (1.9) (see the proof of Claim (1) in (1.10)). Thus, by (1.2) applied to the cycles

g_{1}^{∗}z × Y × Z × □^{m+ℓ} , X × w × Z × □^{n+ℓ} , ∂_{J}W × □^{n′+m′} ,

one has (1.13.2).

Assume now F = 𝔸^{1} × □^{n′+m′} and let us show

ρ_{1}^{∗}z × w × Z × □^{ℓ} P ⌢ ∂_{J}W × 𝔸^{1} × □^{n′+m′} in X × Y × Z × 𝔸^{1} × □^{n+m+ℓ} .

We apply (1.9), Remark, to the ambient variety X × Y × Z × 𝔸^{1} × □^{n+m+ℓ}, the subvarieties

A = z × w × Z × 𝔸^{1} × □^{ℓ} , B = ∂_{J}W × 𝔸^{1} × □^{n′+m′}

and the map ψ∘pr_{𝔸^{1}} : X × Y × Z × 𝔸^{1} × □^{n+m+ℓ} → 𝔸^{1} → G. We have σ(G × A) = X × pr(z) × w × Z × 𝔸^{1} × □^{ℓ}. The subvarieties

X × pr(z) × Y × Z × 𝔸^{1} × □^{m+ℓ} , X × w × Z × 𝔸^{1} × □^{n+ℓ} , ∂_{J}W × 𝔸^{1} × □^{n′+m′}

pairwise meet properly, so by (1.2),

X × pr(z) × w × Z × 𝔸^{1} × □^{ℓ} P ⌢ ∂_{J}W × 𝔸^{1} × □^{n′+m′}

⇔X × pr(z) × Y × Z × 𝔸^{1} × □^{m+ℓ} P ⌢ (X × w × Z × 𝔸^{1} × □^{n+ℓ}) · (∂_{J}W × 𝔸^{1} × □^{n′+m′})
⇔ pr(z) P ⌢ □^{n′} in □^{n}
⇐ z P ⌢ X × □^{n′} in X × □^{n} .

In order to show g_{1}^{∗}z ⊗ ρ_{2}^{∗}w is in C(X_{K}, Y_{E})_{W} we apply the above argument with respect to the variety Y_{K}. We are done with the proof of the special case.

We next assume X general and Y = ℙ_{k}^{N′}. With the notation in (1.12), consider the maps

\frac{Z(X) ⊗ Z(Y)}{[Z(X) ⊗ Z(Y)]_{W}} ⟶^{C_{L_{1}} ⊗id} \frac{Z_{X}(ℙ_{K_{1}}^{N}) ⊗ Z(Y)}{[Z_{X}(ℙ_{K_{1}}^{N}) ⊗ Z(Y)]_{W}} ⟶^{j^{∗}⊗id} \frac{Z(X_{K_{1}}) ⊗ Z(Y)}{[Z(X_{K_{1}}) ⊗ Z(Y)]_{W}} .

The subcomplex [Z_{X}(ℙ_{K_{1}}^{N}) ⊗ Z(Y)]_{W} := [Z(ℙ_{K_{1}}^{N}) ⊗ Z(Y)]_{W} ∩ Z_{X}(ℙ_{K_{1}}^{N}) ⊗ Z(Y) is distinguished. As we have shown the complex in the middle is acyclic. So j^{∗}∘C_{L_{1}} ⊗ id induces zero on homology. Thus π_{1}^{∗} ⊗ 1 and σ_{L_{1}} ⊗ 1 induce the same map on homology. It follows the maps

π^{∗} ⊗ 1 and σ_{L_{r}} ⋯ σ_{L_{1}} ⊗ 1 : \frac{Z(X) ⊗ Z(Y)}{[Z(X) ⊗ Z(Y)]_{W}} → \frac{Z(X_{K_{1}}) ⊗ Z(Y)}{[Z(X_{K_{1}}) ⊗ Z(Y)]_{W}}

induce the same on homology.

We claim σ_{L_{r}} ⋯ σ_{L_{1}} ⊗ 1 induces zero on homology. To show it we may restrict the map to a quasi-isomorphic subcomplex

\frac{Z(X) ⊗ Z_{W} (Y)}{[Z(X) ⊗ Z(Y)]_{W}} ⊂ \frac{Z(X) ⊗ Z(Y)}{[Z(X) ⊗ Z(Y)]_{W}}

and reduce to prove:

Claim. σ_{L_{r}} ⋯ σ_{L_{1}} ⊗ 1 takes Z(X) ⊗ Z_{W} (Y) into [Z(X_{K}) ⊗ Z(Y)]_{W}.

Write σ_{L} for σ_{L_{r}} ⋯ σ_{L_{1}}. Take z ⊗ w ∈ Z(X) ⊗ Z_{W} (Y) with z, w irreducible, and must verify

σ_{L}(z) × w × Z × □^{ℓ} P ⌢ ∂_{J}W × □^{n′+m′} in X × Y × Z × □^{n+m+ℓ} .

The triple of cycles

σ_{L}(z) × Y × Z × □^{m+ℓ} , X × w × Z × □^{n+ℓ} , ∂_{J}W × □^{n′+m′}

pairwise meet properly: the first and third by (1.11), Corollary, and the second and third by w ∈ Z_{W} (Y). The assertion above is equivalent to

σ_{L}(z) × Y × Z × □^{m+ℓ} P ⌢ (X × w × Z × □^{n+ℓ}) · (∂_{J}W × □^{n′+m′}) ,

which again follows from (1.11), Corollary.

If X and Y are general, we repeat the above argument, now that we know the complex Z_{X}(ℙ_{K}^{N}) ⊗ Z(Y)/[Z_{X}(ℙ_{K}^{N}) ⊗ Z(Y)]_{W} is acyclic.

§2. Grothendieck’s construction.

In [Ma], the construction of the category of pure motives is presented. One starts with the correspondence category of smooth projective varieties, takes its pseudo-abelianization, and then add Tate twists.

In this section we construct the same category as follows: take the correspondence category, add Tate twists and then do the pseudo-abelianization. The motivation of the introduction of the correspondence category with Tate twists comes from the construction of D(k).

For an equi-dimensional smooth projective variety X over k, let C^{r}(X) be the ℚ-vector space of ℚ-algebraic cycles of codimension r on X modulo an adequate equivalence relation. (We will be concerned only with rational, homological or numerical equivalence.) An element of C^{∗}(X×Y) is called a C-correspondence from X to Y. One has the operations of transpose, internal product, external product, the induced map by an element of C^{∗}(X × Y) and composition. The composition of correspondences is the map

C^{dim X+a}(X × Y) ⊗ C^{dim Y +b}(Y × Z) → C^{dim X+a+b}(X × Z)

which sends u ⊗ v to

v∘u = (p_{XZ})_{∗}(p_{XY}^{∗} u · p_{Y Z}^{∗}v) .

Here p_{XY}, for example, is the projection X × Y × Z → X × Y.

(2.1) The additive category of finite C-symbols C Symb_{finite}(k) is defined as follows. The objects are of the form

⨁_{α∈A}(X_{α}, r_{α})

(called (finite) C-symbols, or simply symbols ) where A is a finite set, X_{α} is an irreducible smooth projective variety over k and r_{α} ∈ ℤ. (More formally, a symbol is a map from a finite set A into the product of the set of irreducible smooth projective varieties and the set ℤ.) The homomorphism group is the ℚ-vector space

Hom_{CSymb_{finite}(k)}(⊕(X_{α}, r_{α}), ⊕(Y_{β}, s_{β})) = ⨁_{α,β} C^{dim X_{α}+s_{β}−r_{α}}(X_{α} × Y_{β}) .

The composition of morphisms is induced by the composition of correspondences. The category is additive, with the direct sum

(⨁_{α}(X_{α}, r_{α})) ⊕ (⨁_{β}(Y_{β}, s_{β})) = ⨁_{γ}(Z_{γ}, t_{γ})

where γ varies in A ∐ B, and Z_{γ} = X_{α} or Y_{β}, t_{γ} = r_{α} or s_{β}. There is also the tensor product defined as:

(⨁_{α}(X_{α}, r_{α})) ⊗ (⨁_{β}(Y_{β}, s_{β})) = ⨁_{(α,β)}(X_{α} × Y_{β}, r_{α} + s_{β}) .

This operation of tensor product is associative in the strict sense.

In the case C is the rational Chow group, we denote the corresponding category simply by Symb_{finite}(k).

Let (Smooth P roj./k) be the category of not necessarily connected smooth projective varieties and maps. There is a natural contravariant functor

h : (Smooth P roj./k) → C Symb_{finite}(k)

which sends X to X itself and associates with a map f : X → Y the class of its graph Γ_{f} ∈ ⊕C^{dim Y_{β}}(Y_{β} × X) where Y = ∐ Y_{β} (irreducible components).

(2.2) An additive category C is said to be pseudo-abelian if for any object X in C and a projector P ∈ End(X) (i.e. P ∘P = P), the categorical image Im P exists. One then has

X = Im P ⊕ Im(id_{X} − P) .

If C is an additive category, its pseudo-abelianization is a pseudo-abelian category C ˜ together with an additive functor F : C → C ˜ satisfying the condition: for an additive functor F′ : C → C′ where C′ is pseudo-abelian, there exists a unique additive functor G : C ˜ → C′ such that F′ = G∘F.

The pseudo-abelianization exists and can be constructed as follows. The objects of C ˜ are of the form

(X, P) where P ∈ End(X), P ∘P = P ,

and the homomorphism groups are

Hom_{C}˜ ((X, P), (Y, Q)) = Q∘ Hom_{C}(X, Y)∘P .

Composition of morphisms in C ˜ is induced by the composition in C in the obvious manner. The functor F : C → C ˜ is given by X ↦ (X, id_{X}) and the evident isomorphism

Hom_{C}(X, Y) = Hom_{C}˜ ((X, id), (Y, id)) .

One can show that F : C → C ˜ is the pseudo-abelianization; the object (X, P) is the image of P : X → X in C ˜ .

The following property of pseudo-abelianization is easy to see. If C is a tensor category, then C ˜ has a unique structure of a tensor category such that F is a tensor functor. Explicitly, the operation is given by:

(X, P) ⊗ (Y, Q) = (X ⊗ Y, P ⊗ Q) .

(2.3) Definition. The category CM(k) of C-motives is the pseudo-abelianization of the additive category C Symb_{finite}(k). Explicitly, CM(k) has objects

(⨁_{α∈A}(X_{α}, r_{α}), P )

where A is a finite set, X_{α} is irreducible smooth projective, r_{α} ∈ ℤ, and

P = (P_{α β}) ∈ ⨁_{α β} C^{dim X_{α}+r_{β}−r_{α}}(X_{α} × X_{β}) such that P ∘P = P .

The homomorphism groups are

\begin{matrix}Hom( (⊕(X_{α}, r_{α}), P), (⊕(Y_{β}, s_{β}), Q) ) \\ = & Q∘(⨁_{α,β} C^{dim X_{α}+s_{β}−r_{α}}(X_{α} × Y_{β}))∘P ; \end{matrix}

the composition of maps are given by composition of correspondences.

The category is additive and has tensor product

(⨁_{α∈A}(X_{α}, r_{α}), P ) ⊗ (⨁_{β∈B}(Y_{β}, s_{β}), Q) = (⨁_{(α β)}(X_{α} × Y_{β} r_{α} + s_{β}), P ⊗ Q) .

Let ℙ^{1} be the projective line over k, and Π^{2} = ℙ^{1} × p be a cycle on ℙ^{1} × ℙ^{1}, where p is a k-rational point of ℙ^{1}. Denote its cycle class by the same Π^{2} ∈ C^{1}(ℙ^{1} × ℙ^{1}). Then it is a projector, and one has an isomorphism

(pt, 1) →^{∼} (ℙ^{1}, Π^{2}) in CM (k).

Hence

(X,) ≅ X ⊗ (ℙ^{1}, Π^{2})^{⊗r} ≅ (X ⊗ (ℙ^{1})^{r}, Δ_{X} × \overbrace{Π^{2} × ⋯ × Π^{2}}^{r}) .

This shows the category CM_{inf}(k) allows the following description as is in the literature. The objects are of the form (X, P, r) where X is a not necessarily connected smooth projective variety with irreducible components X_{α}, and P = (P_{α β}) ∈ ⊕_{i}C^{dim X_{α}}(X_{α} × X_{β}). The homomorphism group is given by

\begin{matrix}Hom( (X, P, r), (Y, Q, s)) \\ = Q∘(⊕_{α}C^{dim X_{α}+s−r}(X_{α} × Y))∘P ; \end{matrix}

the composition is induced from the composition of correspondences. The object (X, r, P) is the image of the endomorphism of (X, r) given by P.

When C^{∗} is the rational Chow group (resp. algebraic cycles modulo numerical equivalence), the resulting category we denote by CHM(k) and call the category of Chow motives (resp. M(k), the category of Grothendieck motives ). There is a canonical functor cano : CHM(k) → M(k). We summarize:

(Smooth P roj./k) → Symb_{finite} (k) ↪ CHM(k) →^{cano} M(k).

(2.4) Infinite Grothendieck constructions. There is an “infinite” version of construction. The reader may skip this until it is needed in §4.

We first define the additive category of infinite C-symbols C Symb_{inf}(k); it has objects – (infinite) C-symbols:

⨁_{α∈A}(X_{α}, r_{α})

where A is a possibly infinite set, X_{α} is an irreducible smooth projective variety over k and r_{α} ∈ ℤ. We impose that the set {r_{α}} is finite. The homomorphism group is the ℚ-vector space

Hom_{CSymb_{inf} (k)}(⊕(X_{α}, r_{α}), ⊕(Y_{β}, s_{β})) = ∏_{α} ⨁_{β} C^{dim X_{α}+s_{β}−r_{α}}(X_{α} × Y_{β}) ,

i.e., the set of elements u_{α β} such that for each α there are only finitely many non-zero u_{α β}’s. The composition of maps is given by:

(v∘u)_{α β} = ∑_{γ} v_{γ β}∘u_{α γ} .

(Note this is a finite sum.)

This category is additive and has tensor product, the structures being described just as for finite symbols except that one now allows infinite formal sums. If C is the rational Chow group, we denote the category simply by Symb_{inf}(k).

Then we define the category of C-motives of (possibly) infinite type, CM_{inf}(k) to be the pseudo-abelianazation of the category C Symb_{inf}(k). Specifically, it has objects

(⨁_{α∈A}(X_{α}, r_{α}), P )

where

(i) A is a possibly infinite set, X_{α} is irreducible smooth projective, r_{α} ∈ ℤ, such that {r_{α}} is finite, and

(ii) P = (P_{α β}) ∈ ∏_{α} ⨁_{β} C^{dim X_{α}+s_{β}−r_{α}}(X_{α} × Y_{β}) such that P ∘P = P .

The homomorphism groups are

\begin{matrix}Hom((⊕(X_{α}, r_{α}), P)), (⊕(Y_{β}, s_{β}), Q)) \\ = Q∘(∏_{α} ⨁_{β} C^{dim X_{α}+s_{β}−r_{α}}(X_{α} × Y_{β}))∘P ; \end{matrix}

the composition of maps are given by composition of correspondences.

There are full embeddings of categories CM(k) → CM_{inf}(k), and CM_{inf}(k) → Ind CM(k), the latter the ind-category of CM(k).

One has the commutative square

CHM(k) ↪ CHM_{inf}(k)
↓  ↓
M(k) ↪  M(k)

with horizontal arrows full embeddings and vertical ones canonical functors.

§3. C-complexes.

We consider cohomological complexes of abelian groups (or more generally, complexes in an abelian category). We define the notion of C-complexes, which is one possible generalization of that of double complexes. C-complexes are used in constructing the cycle complex of a diagram in §4 and the realization of a motive in §5. For generalities on localizations, triangulated categories and derived categories, we refer to [Har], [Ve].

(3.1) Definition. A C-complex (of abelian groups) consists of

(i) Complexes A^{m} = (A^{m,•}, d_{A^{m}}) (each of which is not necessarily bounded) for m ∈ ℤ, such that for all but finitely many m’s, A^{m} = 0; and

(ii) For m < n, maps of graded groups

F^{m,n} : A^{m,•} → A^{n,•−(n−m−1)}

subject to the condition

F^{m,n}∘(−1)^{m} d_{A^{m}} + (−1)^{n} d_{A^{n}} ∘F^{m,n} + ∑_{m<ℓ<n} F^{ℓ,n}∘F^{m,ℓ} = 0

as a map A^{m,•} → A^{n,•−n+m+2}.

Note that the condition for n = m + 1 amounts to F^{m,m+1} : (A^{m}, d_{A^{m}}) → (A^{m+1}, d_{A^{m+1}}) being a map of complexes. Remark one may put F^{m,m} = (−1)^{m}d_{A^{m}} so that the condition above and the ones below are simplified. We will, however, separate (−1)^{m}d_{A^{m}} and F^{m,n} (m < n) in notation for the purposes in later sections.

A C-complex is denoted by A = (A^{m,•}, d_{A^{m}}, F^{m,n}) = (A^{m}, F^{m,n}). One can associate with it a complex, the total complex T ot (A) = (T ot (A)^{•}, d), defined as:

T ot (A)^{p} = ⨁_{p∈ℤ} A^{m,p−m}

(which is a finite sum), and the differential d is the direct sum, for m, of (−1)^{m} d_{A^{m}} +F^{m,n} : A^{m,•} → ⊕_{n≥m}A^{n,•−(n−m−1)}. The condition in (ii) is equivalent to d being a differential.

One can motivate the definition as follows. The condition for n = m + 1 is equivalent to the map F^{m,m+1} : (A^{m}, d_{A^{m}}) → (A^{m+1}, d_{A^{m+1}}) being a map of complexes. The composition of successive maps F^{m+1,m+2}∘F^{m,m+1} : (A^{m}, (−1)^{m}d_{A^{m}}) → (A^{m+2}, (−1)^{m+2}d_{A^{m+2}}) is required to be homotopic to zero (if not zero as a map of complexes), with the choice of a homotopy −F^{m,m+2}. In the next stage, the map

F^{m+1,m+3}∘F^{m,m+1} + F^{m+2,m+3}∘F^{m,m+2} : (A^{m}, (−1)^{m}d_{A^{m}}) → (A^{m+3}, (−1)^{m+3}d_{A^{m+3}})

is required to be homotopic zero, with the choice of a homotopy −F^{m,m+3}, and so on.

(3.2) Definition. Let B = (B^{m,•}, d_{B^{m}}, G^{m,n}) be another C-complex. A map of Ccomplexes A → B is defined to be a collection maps of graded groups, given for all pairs (m, n):

U^{m,n} : A^{m,•} → B^{n,•−(n−m)}

subject to either of the equivalent conditions:

(i) If tot (U^{m,n}) : T ot (A)^{•} → T ot (B)^{•} is the direct sum of the maps U^{m,n}, then it is a map of complexes.

(ii) For all m, n, one has

(−1)^{n} d_{B^{n}} ∘U^{m,n} + ∑_{ℓ<n} G^{ℓ,n}∘U^{m,ℓ} = U^{m,n}∘(−1)^{m} d_{A^{m}} + ∑_{m<ℓ} U^{ℓ,n}∘F^{m,ℓ}

as a map A^{m,•} → B^{n,•−n+m}.

The set of maps is naturally an abelian group.

For maps of C-complexes U = (U^{m,n}) : A → B and V = (V^{m,n}) : B → C, their composition V ∘U is defined by

(V ∘U)^{m,n} = ∑_{ℓ} V^{ℓ,n}∘U^{m,ℓ} .

C-complexes thus form an additive category.

Let (U^{m,n}) and (Ū^{m,n}) be maps A → B. A homotopy between them is a collection of maps, given for all pairs (m, n),

I^{m,n} : A^{m,•} → B^{n,•−(n−m+1)}

satisfying either of the equivalent conditions:

(i) If tot (I^{m,n}) : T ot (A)^{•} → T ot (B)^{•−1} is the direct sum of the maps I^{m,n}, it gives a homotopy between the maps of complexes tot (U^{m,n}) and tot (Ū^{m,n}).

(ii) For all (m, n) one has

\begin{matrix}U^{m,n} − Ū^{m,n} \\ = & ∑_{ℓ<n} G^{ℓ,n}∘I^{m,ℓ} + (−1)^{n} d_{B^{n}} ∘I^{m,n} + ∑_{m<ℓ} I^{ℓ,n}∘F^{m,ℓ} + I^{m,n}∘(−1)^{m}d_{A^{m}} . \end{matrix}

Two maps are homotopy equivalent if there is a homotopy between them. This is an equivalence relation; it is compatible with sums and compositions. One thus has the homotopy category of C-complexes: the objects are C-complexes and morphisms are homotopy equivalence classes of maps.

Denote by C(Ab) the category of complexes and by K(Ab) its homotopy category. Also denote by (C − Ab) the category of C-complexes of abelian groups, and by K(C − Ab) its homotopy category. There is a commutative square of categories and functors:

(C − Ab) → K(C − Ab)
↓  ↓
C(Ab) →  K(Ab)

where the horizontal arrows are the canonical functors and the vertical arrows are T ot. The functors T ot are fully faithful, and in fact equivalences of categories.

We now show that the category K(C − Ab) has the structure of a triangulated category, and that the functor T ot is an exact functor of triangulated categories.

One defines the translation functor by (A^{m}, F^{m,n})[1] = ((A[1])^{m}, (F [1])^{m,n}) where

(A[1])^{m} = (A^{m+1,•}, −d_{A^{m+1}}), and (F [1])^{m,n} = −F^{m+1,n+1} ;

for a map U : (A^{m}, F^{m,n}) → (B^{m}, G^{m,n}) in K(C −Ab) represented by (U^{m,n}), U[1] : A[1] → B[1] is represented by (U[1])^{m,n} = U^{m+1,n+1}. Note T ot (A[1]) = T ot (A)[1] functorially. (One could also define (A^{m}, F^{m,n}){1} = ((A{1})^{m}, (F {1})^{m,n}) where

(A{1})^{m,•} = (A^{m,•+1}, −d_{A^{m}}), and (F {1})^{m,n} = −F^{m+1,n+1} .

There is an evident isomorphism A[1] →^{∼} A{1}. To fix the idea, we will use the former notion A[1] in the following.)

The cone of U, C_{U} = (C_{U}^{m,•}, d_{C_{U}^{m}}, H^{m,n}) is defined as follows. Take N ≥ 0 so that U^{m,n} = 0 for n − m < −N. Let

\begin{aligned}C_{U}^{m,•} & = A^{m+N+1,•+1} ⊕ B^{m,•}; \\ d_{C_{U}^{m}} & = ( \begin{matrix}−(−1)^{m+N+1}d_{A^{m+N+1}} & 0 \\ U^{m+N+1,m} & (−1)^{m}d_{B^{m}} \end{matrix} ); \\ H^{m,n} & = ( \begin{matrix}−F^{m+N+1,n+N+1} & 0 \\ U^{m+N+1,n+1} & G^{m,n} \end{matrix} ) . \end{aligned}

The C_{U} is independent of the choice of N up to natural isomorphism, and

T ot (C_{U}) = C_{tot(U)}

where the latter is the cone of the map of complexes tot(U) : T ot (A) → T ot (B), defined as in [Har]. We have a triangle

A ⟶^{U} B ⟶ C_{U} ⟶^{[1]} ,

which we declare to be a standard distinguished triangle; upon taking T ot it becomes the standard distinguished triangle in K(Ab) as in [Har]. (We conform to the sign convention in [Har] for distinguished triangles of complexes; the convention of [Ve] is its opposite). Define the class of the distinguished triangles in K(C − Ab) to be the ones isomorphic to the standard distinguished triangles.

One can verify that the translation functor and the distinguished triangles satisfy the axioms of triangulated category. The details of the verification are parallel to those for the category D(k) in §4. So we have the following (not used in this paper).

(3.3) Proposition. There is a natural structure of a triangulated category on K(C − Ab). The functor T ot : K(C − Ab) → K(Ab) is an exact fully faithful functor of triangulated categories.

One can proceed further to discuss the derived category of (C − Ab). (This will not be used in the sequel of this paper.) Consider the class of maps u in K(C − Ab) such that the induced maps tot (u) are quasi-isomorphisms. This satisfies the axioms in [Har] for the category D(C −Ab) obtained by localizing the category K(C −Ab) with respect to this class of maps to have the induced structure of a triangulated category. The functor T ot induces an exact functor (in fact an equivalence) T ot : D(C − Ab) → D(Ab). The following diagram summarizes the constructions in this section.

(C − Ab) → K(C − Ab) → D(C − Ab)
↓ T ot  ↓ T ot  ↓ T ot
C(Ab) →  K(Ab)  →  D(Ab)

§4. The category D(k).

We will define the three triangulated categories D(k), D_{finite}(k), and D_{inf}(k). The category D(k) is called the triangulated category of mixed motives. We first construct the category D_{finite}(k) of finite diagrams and the larger category D_{inf}(k) of (possibly) infinite diagrams. Then we construct D(k) as an intermediate triangulated category:

D_{finite}(k) ↪ D(k) ↪ D_{inf}(k) .

(4.1) In this section, X, Y, ⋯ will denote smooth projective varieties over k. We consider formal symbols of the form

⨁_{α∈I}(X_{α}, r_{α})

where X_{α} is an irreducible smooth projective variety, I an index set (possibly infinite) and r_{α} ∈ ℤ; we impose the condition that the set {r_{α}} is finite. We call such a symbol infinite (resp. finite) if I is infinite (resp. finite). We write 0 for the corresponding symbol when I is an empty set. If X is a not necessarily connected smooth projective variety with irreducible components X_{α}, we define (X, r) = ⊕(X_{α}, r).

Formal symbols were also considered in §2 as objects of the category Symb_{inf}(k); in this section they are to be the building blocks of objects in a larger category D_{inf}(k).

Define the dual of a finite symbol by:

(⊕(X_{α}, r_{α}))^{∨} = ⊕(X_{α}, dim X_{α} − r_{α}) .

Tensor product of two (possibly infinite) symbols by

(⊕(X_{α}, r_{α})) ⊗ (⊕(X′_{α′}, r_{α′}) ) = ⊕(X_{α} × X′_{α′}, r_{α} + r′_{α′}) ;

and, when the first object is finite, inner Hom by

\underline{Hom}(⊕(X_{α}, r_{α}), ⊕(X′_{α′}, r_{α′})) = (⊕(X_{α}, r_{α}))^{∨} ⊗ (⊕(X′_{α′}, r_{α′}) ) .

Define the cycle complex of a formal symbol by

Z^{0}(⊕(X_{α}, r_{α}), ·) = ⊕Z^{r_{α}}(X_{α}, ·) .

This is a homological complex, but often regarded as a cohomological complex in the obvious manner. (If C_{•} is a homological complex, C^{•} = C_{−•} is the corresponding cohomological complex.)

Also define the function complex of a pair of formal symbols by (note the cohomological indexing)

Z^{0}(\underline{Hom}(⊕(X_{α}, r_{α}), ⊕(X_{β}, r_{β}) ) )^{•} := ∏_{α} ⨁_{β} Z^{0}( \underline{Hom}((X_{α}, r_{α}), (X_{β}, r_{β})), −•) .

It consists of elements (f_{αβ}) satisfying the condition that for each α, there are only a finite number of β’s such that f_{αβ} are non-zero. An element of this complex may be called a higher correspondence from ⊕(X_{α}, r_{α}) to ⊕(X_{β}, r_{β}). If ⊕_{α} is a finite direct sum, the function complex coincides with the cycle complex of \underline{Hom}(⊕(X_{α}, r_{α}), ⊕(X_{β}, r_{β}) ); in general, \underline{Hom}(−, −) is only a notation.

There are several operations related to the cycle complexes and function complexes of formal symbols. One may for instance consider the composition of correspondences as follows. There is a partially defined map

Z^{0}(\underline{Hom}(⊕(X_{α}, r_{α}), ⊕(X_{γ}, r_{γ}) ))^{•} ⊗ Z^{0}(\underline{Hom}(⊕(X_{γ}, r_{γ}), ⊕(X_{β}, r_{β}) ))^{•}
−− → Z^{0}(\underline{Hom}( ⊕(X_{α}, r_{α}), ⊕(X_{β}, r_{β}) ))^{•}

given by the composition of correspondences in §1, extended by linearity: f ⊗ g is sent to g∘f where

(g∘f)_{αβ} = ∑_{γ} g_{γβ}∘f_{αγ} (finite sum).

We list below similar operations.

Operations on cycles complexes and function complexes; case of formal symbols. Let K, L, ⋯ be formal symbols.

The map induced by a correspondence. For an element u ∈ Z^{0}(\underline{Hom}(K, L) )^{−ℓ}, the partially defined map (of graded vector spaces)

u_{∗} : Z^{0}(K, ·) − − → Z^{0}(L, · + ℓ), .

One has (∂u)_{∗} = ∂∘u_{∗} − (−1)^{ℓ−1}u_{∗}∘∂.

Transpose. The map

Z^{0}(\underline{Hom}(K, L) )^{•} → Z^{0}(\underline{Hom}(L^{∨}, K^{∨}) )^{•} , u ↦ ^{t}u .

Composition of correspondences. The partially defined map

Z^{0}(\underline{Hom}(K, L))^{•} ⊗ Z^{0}(\underline{Hom}(L, M))^{•} − − → Z^{0}(\underline{Hom}(K, M))^{•} , u ⊗ v ↦ v∘u .

This is associative and satisfies Leibniz formula; it satisfies

t(u∘v) = (−1)^{deg u deg v} ^{t}v∘^{t}u .

External product of cycles. The map

× : Z^{0}(K, ·) ⊗ Z^{0}(L, ·) → Z^{0}(K ⊗ L, ·) , z × w ↦ z × w .

It is associative, satisfies Leibniz formula, and one has

t(z × w) = (−1)^{deg z deg w} ^{t}w ×^{t}z .

Tensor product of correspondences. The map

⊗ : Z^{0}(\underline{Hom}(K, L))^{•} ⊗ Z^{0}(\underline{Hom}(M, N))^{•} → Z^{0}(\underline{Hom}(K ⊗ M, L ⊗ N))^{•} ,

which sends u ⊗ v ∈ Z^{0}(\underline{Hom}(K, L), ℓ) ⊗ Z^{0}(\underline{Hom}(M, N), m) to u ⊗ v ∈ Z^{0}(\underline{Hom}(K ⊗ M, L ⊗ N), ℓ + m) . It is associative, satisfies Leibniz formula, and one has

t(u ⊗ v) = (−1)^{deg u deg v} ^{t}v ⊗^{t}u .

Relations among the operations.

v_{∗}∘u_{∗} = (v∘u)_{∗} .
v∘u = (1 ⊗ v)_{∗}u = (−1)^{deg u deg v}(^{t}u ⊗ 1)_{∗}v (note the sign).
(u∘u′) ⊗ v = (u ⊗ v)∘(u′ ⊗ v); similarly for v .

(4.2) Definition. An object of D_{inf}(k) (resp. D_{finite}(k) ) is a set of data – called an infinite diagram (resp. finite diagram)

K = (K^{m}) = (K^{m}; f^{m,n})

where

(i) For each integer m, K^{m} = ⊕_{α∈I(m)}(X_{α}, r_{α}), a possibly infinite formal symbol (resp. finite symbol). We require: For all but finitely many m’s, K^{m} = 0.

(ii) For (m, n) with m < n, given f^{m,n} = (f_{αβ}^{m,n}) ∈ Z^{0}(\underline{Hom}(K^{m}, K^{n}) )^{−n+m+1}, which are subject to the conditions:

For f^{m_{k},m_{k+1}} ∈ Z^{0}(\underline{Hom}(K^{m_{k}}, K^{m_{k+1}}) )^{−m_{k+1}+m_{k}+1} (k = 1, 2, ⋯, r) one has

f^{m_{r},m_{r+1}} ∘f^{m_{r},m_{r−1}} ∘ ⋯ ∘f^{m_{1},m_{2}} is defined and ∈ Z^{0}(\underline{Hom}(K^{m_{1}}, K^{m_{r+1}}) )^{−m_{r+1}+m_{1}+r} ;

(see Remark to (1.5) for the definition).

For m < n, one has

(−1)^{n} ∂f^{m,n} + ∑_{m<ℓ<n} f^{ℓ,n}∘f^{m,ℓ} = 0 .

Note that the condition includes ∂f^{m,m+1} = 0 (vacuous).

The object with all K^{m} = 0 is denoted by 0. The object with K^{0} = pt, K^{m} = 0 for m ≠ 0 is denoted by \underline{ℚ}.

For an integer r, let the Tate objects be

\underline{ℚ}(r) := (pt, r)[−2r]

namely (pt, r) placed in degree 2r. (\underline{ℚ}(0) = \underline{ℚ}.)

We extend the definitions of cycle complexes and function complexes to diagrams. First, to an object K of D_{inf}(k) we associate its cycle complex Z^{0}(K, ·).

To construct Z^{0}(K, ·), take a distinguished subcomplex Z^{0}(K_{α}^{m}, ·)′ ⊂ Z^{0}(K_{α}^{m}, ·) for each α ∈ I(m) so that the maps

(f_{α β}^{m,n})_{∗} : Z^{0}(K_{α}^{m}, ·)′ → Z^{0}(K_{β}^{n}, · + n − m − 1)′

are all defined. This is possible by Remark to (1.5) and since for each β there are only a finite number of non-zero f_{α β}^{m,n}’s. Let Z^{0}(K^{m}, ·)′ = ⊕Z^{0}(K_{α}^{m}, ·)′. We then have

f_{∗}^{m,n} : Z^{0}(K^{m}, ·)′ → Z^{0}(K^{n}, · + n − m − 1)′ .

The condition (ii) of Definition (4.2) and the formulas after (1.1) imply

(−1)^{n}∂∘f_{∗}^{m,n} + (−1)^{m} f_{∗}^{m,n}∘∂ + ∑ f_{∗}^{ℓ,n}∘f_{∗}^{m,ℓ} = 0 ,

namely (Z^{0}(K^{m}, ·), f_{∗}^{m,n}) is a C-complex. We define Z^{0}(K, ·) to be its total complex T ot (Z^{0}(K^{m}, ·), f_{∗}^{m,n}), i.e. the complex (𝕂, d) with

𝕂^{i} = ⨁_{j} Z^{0}(K^{j}, j − i)′ ,

and

d^{i} = ∑_{j}((−1)^{j}∂_{j} + ∑_{j<ℓ} f_{∗}^{j,ℓ}) .

The result is unique up to canonical quasi-isomorphism: Denote temporarily by Z^{0}(K, ·)′ the total complex above, and by Z^{0}(K, ·)′′ the total complex coming from another choice of distinguished subcomplexes. Then the two inclusions

Z^{0}(K, ·)′ ← Z^{0}(K, ·)′ ∩ Z^{0}(K, ·)′′ → Z^{0}(K, ·)′′

are both quasi-isomorphisms.

Let (K; f) and (L; g) be objects in D_{inf}(k). The function complex Hom(K, L)^{•} is a cochain complex of abelian groups defined as follows. (If K = (pt, 0), it coincides with the cycle complex of L.) By an argument similar to the one above, one can take distinguished subcomplexes

Z^{0}(\underline{Hom}(K_{α}^{m}, L_{β}^{m′}) )′^{•} ⊂ Z^{0}(\underline{Hom}(K_{α}^{m}, L_{β}^{m′}) )^{•}

so that if

Z^{0}(\underline{Hom}(K^{m}, L^{m′}) )′^{•} := ∏_{α} ⨁_{β} Z^{0}(\underline{Hom}(K_{α}^{m}, L_{β}^{m′}) )′^{•}

the maps

(−)∘f^{n,m} : Z^{0}(\underline{Hom}(K^{m}, L^{m′}) )′^{•} → Z^{0}(\underline{Hom}(K^{n}, L^{m′}) )′^{•−n+m+1}

are defined; similarly for g^{m′,n′} ∘(−). In the following we omit the ′ which indicates subcomplex for simplicity. The cohomological complex to be defined has the group of N-cochains

Hom(K, L)^{N} = ⨁_{−m+m′+p=N} Z^{0}(\underline{Hom}(K^{m}, L^{m′}) )^{p}

The differential of this complex, which we denote by D, is the sum of the three types of maps:

(−1)^{p+m′+n+1} (∘ f^{n,m}) : Z^{0}(\underline{Hom}(K^{m}, L^{m′}) )^{p} → Z^{0}(\underline{Hom}(K^{n}, L^{m′}) )^{p−n+m+1} ,
(−1)^{m′+n′} (g^{n′,m′} ∘) : Z^{0}(\underline{Hom}(K^{m}, L^{m′} )^{p} → Z^{0}(\underline{Hom}(K^{m}, L^{n′}))^{p−n′+m′+1} ,

and

(−1)^{m′} ∂ : Z^{0}(\underline{Hom}(K^{m}, L^{m′}) )^{•} → Z^{0}(\underline{Hom}(K^{m}, L^{m′}) )^{•+1} .

One verifies D∘D = 0. The complex is defined uniquely up to canonical quasi-isomorphism ( as for Z^{0}(K, ·)). An element of this complex is a higher correspondence from K to L.

If K happens to be finite, so that \underline{Hom}(K, L) exists as an object (to be defined in (4.5) ), its cycle complex Z^{0}(\underline{Hom}(K, L), ·) and the complex defined above are the same up to canonical quasi-isomorphism.

As for formal symbols one may define operations on the cycle and function complexes of diagrams; we first need to consider the composition of correspondences. Given three objects K, L and M, the composition is the partially defined map

Hom(K, L)^{•} ⊗ Hom(L, M)^{•} − − → Hom(K, M)^{•}

u ⊗ v ↦ v∘u; (v∘u)^{m,n} = ∑_{ℓ∈ℤ} v^{ℓ,n}∘u^{m,ℓ} .

It is associative and satisfies the Leibniz formula

D(v∘u) = Dv∘u + (−1)^{deg v}v∘Du ,

where deg v is the total degree of v in the cohomological complex.

(4.3) Proposition. There is a quasi-isomorphic subcomplex, unique up to canonical quasiisomorphism, of the tensor product

Hom(K, L)^{•} ⊗ Hom(L, M)^{•}

on which the composition u ⊗ v ↦ v∘u ∈ Hom(K, M)^{•} is defined.

Proof. In each double complex

C^{m,ℓ,n} := Z^{0}(\underline{Hom}(K^{m}, L^{ℓ}) )^{•} ⊗ Z^{0}(\underline{Hom}(L^{ℓ}, M^{n}) )^{•}

one can take a distinguished subcomplex (C^{m,ℓ,n})′ on which the composition map into Z^{0}(\underline{Hom}(K^{m}, M^{n}) )^{•} is defined, (1.7) (3). Then those subcomplexes (C^{m,ℓ,n})′ form a quasiisomorphic subcomplex C′ of the double complex considered in the statement.

By definition

Hom_{D_{inf} (k)}(K, L) = H^{0}Z^{0}(\underline{Hom}(K, L) )^{•} .

The composition of morphisms is induced from the composition of the function complexes. A morphism u : K → L is represented by u^{m,n} ∈ Z^{0}(\underline{Hom}(K^{m}, L^{n}), n − m)′ (for a choice of a collection of distinguished subcomplexes) subject to the condition

(−1)^{n}∂u^{m,n} − ∑(−1)^{m+ℓ} u^{ℓ,n}∘f^{m,ℓ} + ∑(−1)^{ℓ+n} g^{ℓ,n}∘u^{m,ℓ} = 0 .

It defines the zero morphism if there exist U^{m,n} ∈ Z^{0}(\underline{Hom}(K^{m}, L^{n}), n − m + 1)′ such that

u^{m,n} = (−1)^{n}∂U^{m,n} + ∑(−1)^{m+ℓ} U^{ℓ,n}∘f^{m,ℓ} + ∑(−1)^{ℓ+n} g^{ℓ,n}∘U^{m,ℓ} .

(4.4) Proposition. (1) The category D_{inf}(k) is an additive category.

(2) The association K ↦ Z^{0}(K, ·) is a functor D_{inf}(k) → D(ℚ), the latter the unbounded derived category of ℚ-vector spaces.

(3) There is a fully faithful functor Symb_{inf}(k) → D_{inf}(k) such that the composition

(Smooth Proj./k) ^{opp} → Symb_{inf}(k) → D_{inf}(k)

(which will also be denoted h) assigns to a smooth projective variety X the diagram (X, 0)[0] (i.e. (X, 0) placed in degree 0).

All the statements are valid if D_{inf}(k), Symb_{inf}(k), are replaced with D_{finite}(k) and Symb_{finite}(k), respectively.

Proof. (1) is easily seen. For (2), let (u^{m,n}) represent a morphism u : K → L; then (u_{∗}^{m,n}) satisfy the condition

(−1)^{n}∂∘u_{∗}^{m,n} + ∑_{ℓ} g_{∗}^{ℓ,n}∘u_{∗}^{m,ℓ} = (−1)^{m}u_{∗}^{m,n}∘∂ + ∑_{ℓ} u_{∗}^{ℓ,n}∘f_{∗}^{m,ℓ} .

So (u_{∗}^{m,n}) represents a map of C-complexes

(Z^{0}(K^{m}, ·), f_{∗}^{m,n}) → (Z^{0}(L^{m}, ·), g_{∗}^{m,n})

and induces a map

tot (u_{∗}^{m,n}) : Z^{0}(K, ·) → Z^{0}(L, ·) .

If (U^{m,n}) gives (u^{m,n}) ≃ 0, then tot (U_{∗}^{m,n}) gives tot (u_{∗}^{m,n}) ≃ 0.

(3) The functor sends an object ⊕(X_{α}, r_{α}) to ⊕(X_{α}, r_{α}) itself placed in degree 0 (i.e. the object K such that K^{0} = ⊕(X_{α}, r_{α}), K^{m} = 0 for m ≠ 0). The corresponding homomorphism groups are identical.

(4.5) Duals, tensor products, and inner Hom’s. One can define the dual K^{∨} and tensor product K ⊗ L as follows. For K finite, define the dual K^{∨} = ((K^{∨})^{m}, (f^{∨})^{m,n}) by (K^{∨})^{−m} = (K^{m})^{∨} = ⊕_{α∈I(m)}(K_{α}^{m})^{∨}, and, for m < n, α ∈ I(m), β ∈ I(n),

\begin{matrix}(f^{∨})^{−n,−m} = (−1)^{mn+m+1} ^{t}f^{m,n} ∈ & Z^{0}(\underline{Hom}(K^{−n ∨}, K^{−m ∨}))^{n−m−1} \\ = & Z^{0}(\underline{Hom}(K^{m}, K^{n}))^{n−m−1} . \end{matrix}

The association K ↦ K^{∨} is functorial. In fact, define an isomorphism of complexes

α(K, L) : Hom(K, L)^{•} → Hom(L^{∨}, K^{∨})^{•}

by

\begin{matrix}Z^{0}(\underline{Hom}(K^{m}, L^{m′}) )^{p} → Z^{0}(\underline{Hom}((L^{∨})^{−m′}, (K^{∨})^{−m}) )^{p} \\ u ↦ u^{∨} := (−1)^{mm′+mp+m′p+m+1} ^{t}u . \end{matrix}

(Note for f^{m,n} ∈ Z^{0}(\underline{Hom}(K^{m}, K^{n}) )^{−n+m+1}, (f^{∨})^{−n,−m} coincides with (f^{m,n})^{∨}.) One verifies the transitivity: if u ∈ Hom(K, L)^{0} and v ∈ Hom(L^{∨}, K^{∨})^{0}, then (v∘u)^{∨} = u^{∨}∘v^{∨}. So the α (K, L) induces an isomorphism Hom (K, L) →^{∼} Hom (L^{∨}, K^{∨}), also satisfying the transitivity. Hence the association K → K^{∨} is a functor.

There is an isomorphism ι_{K} : K →^{∼} K^{∨ ∨} given by (−1)^{m} on K^{m} → (K^{∨ ∨})^{m} = K^{m}. It is shown that the composition

α(L^{∨}, K^{∨})∘α(K, L) : Hom(K, L)^{•} → Hom(L^{∨}, K^{∨})^{•}
→ Hom(K^{∨ ∨}, L^{∨ ∨})^{•} →^{∼} Hom(K, L)^{•} ,

where the last isomorphism is the one induced by ι_{K} and ι_{L}, is the identity. So for a morphism u : K → L the square

K ⟶^{ι_{K}} K^{∨ ∨}
u↓  ↓_{u^{∨ ∨}}
L ⟶^{ι_{L}} L^{∨ ∨}

commutes.

The tensor product (K; f) ⊗ (L; g) is the object (M; h) where

M m̄ = ⨁_{(α,β)∈I(m)×I(m′),m+m′=m̄} K_{α}^{m} ⊗ L_{β}^{m′} , h^{m̄,n̄} = (h_{(α,β),(α′,β′)}^{m̄,n̄});

h_{(α,β),(γ,β)}^{m+m′,n+m′} = (−1)^{m′(n−m−1)} f_{α,γ}^{m,n} ⊗ id_{L_{β}^{m′}} ,
h_{(α,β),(α,γ)}^{m+m′,m+n′} = (−1)^{m}id_{K_{α}^{m}} ⊗ g_{β,γ}^{m′,n′} ,

and h_{(α,β),(α′,β′)}^{m̄,n̄} = 0 otherwise.

For K of finite type, define the inner Hom by

\underline{Hom}(K, L) = K^{∨} ⊗ L .

Operations on cycles complexes and function complexes; case of diagrams. Let K, L, ⋯ be objects in D_{inf}(k). The differences from the case of formal symbols is: one has the dual in place of the transpose; there are no signs in some formulas involving the duals.

The map induced by a correspondence. For an element u ∈ Hom(K, L)^{−ℓ}, the partially defined map (of graded vector spaces)

u_{∗} : Z^{0}(K, ·) − − → Z^{0}(L, · + ℓ), .

One has (∂u)_{∗} = ∂∘u_{∗} − (−1)^{ℓ−1}u_{∗}∘∂.

Dual. The map (defined above)

∨ : Hom(K, L)^{•} → Hom(L^{∨}, K^{∨})^{•} , u ↦ u^{∨} .

Composition of correspondences. The partially defined map

Hom(K, L)^{•} ⊗ Hom(L, M)^{•} − − → Hom(K, M)^{•} , u ⊗ v ↦ v∘u .

This is associative and satisfies Leibniz formula; (if K, L are symbols) it satisfies

(u∘v)^{∨} = v^{∨}∘u^{∨} ;

unlike the case of formal symbols (where transposes appeared), there is no sign. External product of cycles. The map

˜× : Z^{0}(K, ·) ⊗ Z^{0}(L, ·) → Z^{0}(K ⊗ L, ·)

which sends (z, w) ∈ Z^{0}(K^{m}, p) ⊗ Z^{0}(L^{n}, q) to z ˜×w := (−1)^{np}z × w. It is associative, satisfies Leibniz formula, and one has

(z ˜×w)^{∨} = w^{∨} ˜×z^{∨} .

Tensor product of correspondences. The map

˜⊗ : Hom(K, L)^{•} ⊗ Hom(M, N)^{•} → Hom(K ⊗ M, L ⊗ N)^{•}

which sends (u, v) ∈ Z^{0}(Hom(K^{m}, L^{m′}), p) ⊗ Z^{0}(Hom(M^{n}, N^{n′}), q) to u ˜⊗v := (−1)^{n′p+mq+mn′+nm′} u ⊗ v. It is associative, satisfies Leibniz formula, and one has

(u ˜⊗v)^{∨} = v^{∨} ˜⊗u^{∨} .

Relations among the operations.

v_{∗}∘u_{∗} = (v∘u)_{∗} .
v∘u = (1 ˜⊗v)_{∗}u = (u^{∨} ˜⊗1)_{∗}v .
(u∘u′) ˜⊗v = (u ˜⊗v)∘(u′ ˜⊗v); similarly for v .
(u ˜⊗v)_{∗} = u_{∗} ⊗ v_{∗} .

The above operations on complexes induce maps on cohomology. So we have maps

Hom(K, L) → Hom(L^{∨}, K^{∨}) , u ↦ u^{∨} ,
Hom(K, L) ⊗ Hom(L, M) → Hom(K, M) , u ⊗ v ↦ v∘u ,
Hom(K, L) ⊗ Hom(M, N) → Hom(K ⊗ M, L ⊗ N) , u ⊗ v ↦ u ˜⊗v ,

and relations

(u∘v)^{∨} = v^{∨}∘u^{∨} ,
(u ˜⊗v)^{∨} = v^{∨} ˜⊗u^{∨} ,

and

(u∘u′) ˜⊗v = (u ˜⊗v)∘(u′ ˜⊗v) , u∘(v∘v′) = (u∘v) ˜⊗(u∘v′) .

So the association (K, L) ↦ K ⊗ L is a bifunctor.

As a special case, one has the functor from D_{inf}(k) to itself, called the r-th Tate twist

K ↦ K(r) := K ⊗ \underline{ℚ}(r);

concretely, K(r) = (K(r)^{m}, f(r)^{m,n}) where (K(r))^{m} = K^{m+2r} ⊗ (pt, r) and (f(r))^{m,n} = f^{m+2r,n+2r} ⊗ id_{(pt,r)}.

Remark. For a smooth projective variety X, h(X)(r) is (X, r)[−2r], namely “(X, r) placed in degree 2r”. On the other hand, h( (X, r) ) = (X, r)[0]. In other words, the Tate twists are not compatible with the functor h.

Properties of tensor product and dual.

Associativity. One has (K ⊗ L) ⊗ M = K ⊗ (L ⊗ M). (The objects are identical, not only canonically isomorphic.)

Unit object. ℚ ⊗ K = K ⊗ ℚ = K.

Commutativity. There is a canonical isomorphism σ : K ⊗ L ≅ L ⊗ K. The σ consists of (−1)^{mm′} on K^{m} ⊗ L^{m′} = L^{m′} ⊗ K^{m}.

Tensor product and dual. (K ⊗ L)^{∨} = L^{∨} ⊗ K^{∨}. The objects are identical.

Tensor product and \underline{Hom}. There is a canonical isomorphism

\underline{Hom}(K, \underline{Hom}(L, M)) ≅ \underline{Hom}(K ⊗ L, M)

if K, L are of finite type. Indeed the left hand side equals K^{∨}⊗(L^{∨}⊗M) = (K^{∨}⊗L^{∨})⊗M = (L ⊗ K)^{∨} ⊗ M ≅ (K ⊗ L)^{∨} ⊗ M. Thus one has

Hom(K, \underline{Hom}(L, M)) ≅ Hom(K ⊗ L, M) .

We now describe the structure of a triangulated category on D_{inf}(k). Define the translation functor K ↦ K[1] in the category D_{inf}(k) as follows. On objects (K[1])^{m} = K^{m+1} and (f[1])^{m,n} = (−1)^{m+n+1}f^{m+1,n+1}. For a morphism u : K → L represented by (u^{m,n}), u[1] : K[1] → L[1] is to be represented by (u[1])^{m,n} = (−1)^{m+n}u^{m+1,n+1}.

Define the cone of a morphism u : K → L, represented by (u^{m,n}) to be the object C_{u} = (C; h) where

C^{m} = K^{m+1} ⊕ L^{m}

and

h^{m,n} = ( \begin{matrix}(−1)^{m+n+1}f^{m+1,n+1} & 0 \\ u^{m+1,n} & (−1)^{m+n}g^{m,n} \end{matrix} ) .

There are canonical morphisms α(u) : L → C_{u} given by

α(u)^{m,n} = ( \begin{matrix}0 \\ (−1)^{m}δ_{m,n}id_{L^{m}} \end{matrix} ) : L^{m} → K^{n+1} ⊕ L^{n}

and β(u) : C_{u} → K[1] given by

β(u)^{m,n} = ( δ_{m,n} id_{K^{m+1}} 0 ) : K^{m+1} ⊕ L^{m} → K^{m+1}

so that we have a triangle

K → L → C_{u} →^{[1]} .

Such a triangle is called a standard distinguished triangle. We declare the distinguished triangles in the category D_{inf}(k) to be the ones isomorphic to the standard ones. The category D_{finite}(k) also has parallel structures.

(4.6) Theorem. The category D_{inf}(k) (resp. D_{finite}(k)) equipped with the translation functor and the distinguished triangles is a triangulated category.

Proof. The proof is parallel to the case of the derived category of complexes; see e.g. [KS, §1.4] for a detailed exposition. To illustrate, we show the following, which implies that triangles may be “turned”. (The octahedral axiom is verified in a similar manner.) There exists an isomorphism ϕ : K[1] → C_{α(u)} such that the following diagram commutes:

L ⟶^{α(u)} C_{u} ⟶^{β(u)} K[1] ⟶^{−u[1]} L[1]
id↓  ^{id}↓  ^{ϕ}↓  ^{id}↓
L ⟶^{α(u)} C_{u} ⟶^{α(α(u))} C_{α(u)} ⟶^{β(α(u))} L[1] .

In fact if one takes

ϕ^{m,n} = ( \begin{matrix}(−1)^{m+n+1}u^{m+1,n+1} \\ (−1)^{m}δ_{m,n}id_{K^{m+1}} \\ 0 \end{matrix} ) : K^{m+1} → L^{n+1} ⊕ K^{n+1} ⊕ L^{n}

then

ψ^{m,n} = (0, (−1)^{m}δ_{m,n}id_{K^{m+1}}, 0) : L^{m+1} ⊕ K^{m+1} ⊕ L^{m} → K^{n}

can be checked to be the inverse by a lengthy calculation.

Let K be an object of D_{inf}(k). The object ⨁_{α≥1} K_{α} (K_{α} is a copy of K and α varies over natural numbers) in an obvious manner. It satisfies, for any object L of D_{inf}(k), Hom(⊕K_{α}, L) = ∏ Hom(K_{α}, L) and Hom(L, ⊕K_{α}) = ⊕ Hom(L, K_{α}); namely it is a direct sum and a direct product. (To see these note for example Hom(⊕K_{α}, L)^{•} = ∏ Hom(K_{α}, L)^{•}. )

For an object K in D_{inf}(k) and a projector p : K → K, let [K, p] be the cone of the morphism

u = (u_{α,β}) : ⨁_{α≥1} K_{α} → ⨁_{α≥1} K_{α}

where K_{α} is a copy of K and u_{α,α} = id, u_{α,α+1} = −p, and u_{α,β} = 0 if otherwise. The “ first direct summand” morphism K → ⊕K_{α} followed by the canonical morphism ⊕K_{α} → [K, p] gives a morphism K → [K, p]. It induces canonical functorial maps

Hom(L, K) → Hom(L, [K, p])

and

Hom([K, p], L) → Hom(K, L) .

(4.7) Proposition. The above maps induce isomorphisms

Hom(L, [K, p] ) ≅ p∘ Hom(L, K) ;
Hom([K, p], L) ≅ Hom(K, L)∘p .

There is a canonical isomorphism K = [K, p] ⊕ [K, 1 − p] according to which p = ( \begin{matrix}id & 0 \\ 0 & 0 \end{matrix} ); in particular [K, p] is the image of the projector p.

Proof. One has a long exact sequence

→ Hom(⊕K_{α}, L[i − 1]) → Hom( [K, p], L[i]) → Hom(⊕K_{α}, L[i]) →^{∘u} Hom(⊕K_{α}, L[i]) → .

Note that ∘u is surjective with kernel Hom(K, L[i])∘p, more precisely, the following diagram is commutative:

Ker (∘u)  ↪  Hom(⊕K_{α}, L[i])
↓≅  ↓ 1st projection
Hom(K, L[i])∘p ↪  Hom(K, L[i])

One hence obtains Hom([K, p], L) = Hom(K, L)∘p.

As for Hom(L, [K, p] ) one argues similarly, now using the commutative diagram

Hom(L[i], ⊕K_{α})  ↠  Cok (u∘)
↑ 1st summand  ↑≅
Hom(L[i], K)  ↠ p∘ Hom(L[i], K) .

(4.8) Definition. Let D(k) be the full subcategory of D_{inf}(k) consisting of objects of the form [K, p] with K an object of D_{finite}(k) and p a projector.

This is a triangulated subcategory. For that, we must show the objects [K, p] are closed under shifts and taking cones. The former is obvious by [K, p][1] = [K[1], p[1] ]. For the latter, take a morphism u_{11} : [K, p] → [L, q], its cone is a direct summand of the cone of the

morphism u = ( \begin{matrix}u_{11} & 0 \\ 0 & 0 \end{matrix} ) : K → L, the matrix being with respect to K = [K, p] ⊕ [K, 1 − p]

and L = [L, q] ⊕ [L, 1 − q]. Thus C_{u_{11}} is a direct summand of C_{u}.

The category is closed under the operations of duals, tensor products, and inner Hom’s:

[K, p]^{∨} = [K^{∨}, p^{∨}] and
[K, p] ⊗ [L, q] = [K ⊗ L, p ⊗ q] .

The category of Chow motives is fully embedded in D(k): in fact let the functor be:

(X, r, P) ↦ [(X, r)[0], P]

where the right hand side is the image of the projector P : (X, r)[0] → (X, r)[0]. The homomorphism groups between such objects are identical. Hence we have

(4.9) Theorem. There are fully faithful functors CHM(k) → D(k) and CHM_{inf}(k) → D_{inf}(k).

The following summarizes the main properties of D(k) which can be shown without assuming any conjectures. The properties (2) (4) will be discussed in subsequent sections.

(3) is clear from the definitions and Theorem (1.1).

(4.10) Theorem. The triangulated category D(k) has the following properties:

(1) D(k) has dual, tensor product, inner Hom, the unit object \underline{ℚ}, and the Tate objects \underline{ℚ}(r);

(2) (ch(k) = 0) The contravariant functor h : (Smooth P roj./k) → D(k) can be extended to a functor from the category of quasi-projective varieties; to each quasi-projective X there is associated its (cohomological) motive h(X), an object of D(k);

(3) If X is smooth and projective, one has

Hom_{D(k)}(\underline{ℚ}, h(X)(r)[2r − m]) = K_{m}(X)_{ℚ}^{(r)} .

Here the right side is an Adams-graded piece of the K-group of X.

(4) If H^{∗} denotes either of Betti or etale cohomology, there is the corresponding realization functor H^{∗} : D(k) → V ect (Λ); here V ect (Λ) is the category of Λ-vector spaces, and Λ is the coefficient field of the cohomology theory.

(4.11) Definition. For an object K of D_{inf}(k), define

Z^{r}(K, ·) := Z^{0}(K ⊗ (pt, r), ·)

and the higher Chow group (or motivic cohomology) of K by

CH^{r}(K, n) := H_{n}Z^{r}(K, ·) .

Note CH^{r}(K, n) = Hom_{D_{inf} (k)}(\underline{ℚ}, K(r)[2r − n]).

§5. Cohomology realizations of motives.

The purpose of this section is to construct the realization functors. We benefited from a discussion with S. Bloch and [Bl 4]. We consider Betti and etale cohomology. Let Λ be the coefficient field:

(1) Betti: H^{∗} = H_{B}^{∗}. k ⊂ ℂ and Λ = ℚ.

(2) Etale: H^{∗} = H_{et}^{∗}. k is a separably closed field, and Λ = ℚ_{ℓ} where ℓ is a prime integer not dividing char (k).

Let (Λ − vect) be the category of Λ-vector spaces, and D(Λ − vect) the derived category of complexes of Λ-vector spaces with unbounded cohomology. In each case (1) or (2) there is the cohomology functor ℝΓ : (Smooth P roj./k)^{opp} → D(Λ − vect), and the induced functor H^{∗} : (Smooth P roj./k)^{opp} → (Λ − vect). If k is a field, not necessarily separably closed, one can consider the composition with base change (k̄ is the separable closure)

(Smooth P roj./k)^{opp} → (Smooth P roj./k̄)^{opp} ⟶^{ℝΓ} D(Λ − vect) .

(5.0) Theorem. Let H^{∗} denote either of H_{B}^{∗} or H_{et}^{∗}. There is an exact functor of triangulated categories

ℝΓ : D_{inf}(k) → D (Λ − vect)

the composition of which with h : (Smooth P roj./k)^{opp} → D_{inf}(k) coincides with the cohomology functor X ↦ ℝΓ(X). We have also the induced functor

H^{∗} : D_{inf}(k) → (Λ − vect) .

In the etale case, if k is not separably closed, one also considers the composition of ℝΓ with base change D_{inf}(k) → D_{inf}(k̄) → D (Λ − vect).

In this paper we restrict ourselves to the case of Betti cohomology. The case of etale cohomology will be treated in a separate paper. Let R = ℚ for the rest of this section. For a smooth projective variety X over k ⊂ ℂ, consider sheaves of R-modules on X(ℂ) with usual topology. Let Mod(X, R) be the category of sheaves of R-modules on X, C^{∗}(X, R) the category of chain complexes of R-modules with boundedness condition ∗, and D^{∗}(X, R) the derived category of R-modules.

For an R-module A on X(ℂ) let C^{•}(A) be its Godement resolution; it is a complex of injective sheaves. We recall the definition. Let i_{x} : x → X be the inclusion of a point x. Let

C^{0}(A) := ∏ i_{x ∗}i_{x}^{∗}A

where x runs over the set of points. Then C^{0}(A) is an injective sheaf and there is a canonical injection A → C^{0}(A). Define C^{1}(A) = C^{0}(Cok(A → C^{0}(A))), and

C^{p}(A) = C^{0}(Cok(C^{p−2}(A) → C^{p−1}(A)))

for p ≥ 2. One has an injective resolution A → C^{•}(A).

For simplicity let

C_{X} = C_{X}^{•} := C^{•}(R_{X})

and

K_{X} = K_{X}^{•} := τ_{≤2 dim X}C^{•}(R_{X}) .

These are both resolutions of R_{X} and will play fundamental roles. The former is injective, and the latter is c-soft, namely for any open U ⊂ X(ℂ), H_{c}^{i}(U, K_{X}^{p}) = 0 for i > 0 and each p. (See [Bo, Chap. V].)

Let X and Y be smooth projective varieties over k, and p, q be the projections from X × Y to X and Y, respectively. There is the theory of Verdier duality, in particular the functor f^{!} for a map f. Available references include [Bo], [KS] and [Ve]. The analogous theory in etale topology is developed in [SGA 4, XVIII]. We will give a definition of q^{!} below. We employ the notation and terminology in [Bo].

The complex of sheaves p^{∗}K_{X} is c-soft when restricted to any fiber of q. Thus it can be used to calculate q^{!}. More specifically

K_{q}(B) := p^{∗}K_{X} ⊗ B

is a bounded, fiberwise c-soft resolution of B, and the functor

Mod(X × Y, R) → C^{b}(Y, R)

given by B ↦ q_{∗}K_{q}(B) has a right adjoint

q_{•}^{!} : Mod(Y, R) → C^{b}(X × Y, R)

so that there is an isomorphism of complexes

(*)  Hom(q_{∗}K_{q}(B), A) = Hom(B, q_{•}^{!}A) .

where for A in Mod(Y, R) and B in Mod(X × Y, R).

Concretely, for an open set U ⊂ X × Y, one has

(q_{i}^{!}A)(U) = Hom(q_{∗}K_{q}^{i}(R_{U}), A)

where R_{U} is the extension by zero of R_{X}|_{U}. The q_{•}^{!} extends to a functor

q_{•}^{!} : C(Y, R) → C(X × Y, R) .

The duality (∗) holds for complexes of sheaves A ∈ C(Y, R) and B ∈ C^{b}(X × Y, R) as well. The q_{•}^{!} is left exact, commutes with direct product, and takes injectives to injectives. The functor q_{•}^{!} has right derived functor

q^{!} : D^{+}(Y, R) → D^{+}(X × Y, R)

and we have for A ∈ D(Y, R) and B ∈ D^{b}(X × Y, R)

Hom(ℝq_{∗}(B), A) = Hom(B, q^{!}A) .

However in the sequel we will stick with the q^{!} at the chain level, so we drop the dot from q_{•}^{!} and simply write q^{!}. For A in C(Y, R) one has a canonical map

q_{∗}q^{!}A → A .

The following is a form of Poincaré duality. (More generally it holds for proper smooth maps.)

(5.1) Theorem. For a complex of injective sheaves A in C(Y, R) there is a canonical quasi-isomorphism γ_{q}(A) : q^{∗}A → q^{!}A[−2 dim X].

The stated map is induced from the trace map ℝq_{∗}R → R^{2d}q_{∗}R[−2d] → R[−2d].

If we take A = C_{Y} we have a canonical quasi-isomorphism ρ : R_{X×Y} → q^{∗}C_{Y} ( induced from the canonical map R_{Y} → C_{Y}). Composing with the map γ_{q}(C_{Y}), one has a quasiisomorphism

β : R_{X×Y} ⟶^{ρ} q^{∗}C_{Y} ⟶^{γ_{q}(C_{Y} )} q^{!}C_{Y} [−2 dim X] .

Let Z be another smooth projective variety, r and s the projections from Y × Z to Y and Z, and q′, r′ be projections in the Cartesian diagram:

X × Y × Z
r′ ↙  ↘ q′
X × Y  Y × Z
q ↘  ↙ r
Y

As for the projection q, one defines for q′ the functor K_{q′}(B) := B ⊗ r′^{∗}p^{∗}K_{X} and let q′^{!} be the corresponding adjoint. Similarly, for r, one has K_{r} and r^{!}, and for r′, K_{r′} and r′^{!}.

(5.2) Proposition. (1) For A on Y × Z there is a canonical isomorphism

q^{!}r_{∗}A = r′_{∗}q′^{!}A .

(2) For B on Y there is a canonical map

ϵ : r′^{∗}q^{!}B → q′^{!}r^{∗}B .

(3) For B on Y there is a canonical isomorphism

q′^{!}r^{!}B = r′^{!}q^{!}B .

Proof. (1) For a sheaf B on X × Y, using duality we have

Hom(B, r′_{∗}q′^{!}A) = Hom(q′_{∗}K_{q′}(r′^{∗}B), A)

and

Hom(B, q^{!}r_{∗}A) = Hom(r^{∗}q_{∗}K_{q}(B), A) .

Since r^{∗}q_{∗}K_{q}(B) = q′_{∗}r′^{∗}K_{q}(B) by proper base change, and r′^{∗}K_{q}(B) = K_{q}(r′^{∗}B) by definition, the two are the same.

(2) Applying q^{!} to the canonical map B → r_{∗}r^{∗}B, one has a map q^{!}B → q^{!}r_{∗}r^{∗}B = r′_{∗}q′^{!}r^{∗}B. By adjointness one has a map as stated.

(3) For C on X × Y × Z, one has

Hom(C, q′^{!}r^{!}B) = Hom(r_{∗}(q′_{∗}(C ⊗ r′^{∗}p^{∗}K_{X} ⊗ s^{∗}K_{Z}), B)
= Hom(r_{∗}q′_{∗}(C ⊗ r′^{∗}p^{∗}K_{X} ⊗ q′^{∗}r^{∗}K_{Y}), B) .

One used that f_{∗}(A ⊗ f^{∗}B) = f_{∗}A ⊗ B if B is flat, [Bo, p. 160]. For Hom(C, r′^{!}q^{!}B) one has the same.

We define an object of C(X × Y, R) by

Hom(X, Y) = Hom(p^{∗}C_{X}, q^{!}C_{Y})[−2 dim X] .

There is an isomorphism α_{XY} : R_{X×Y} →^{∼} Hom(X, Y) in D(X×Y, R) obtained by composing the quasi-isomorphisms

R_{X×Y} → q^{!}C_{Y} [−2 dim X] ← Hom(p^{∗}C_{X}, q^{!}C_{Y})[−2 dim X] .

The latter map is the one induced from ρ : R_{X×Y} → p^{∗}C_{X}. Since p^{∗}C_{X} is flat and q^{!}C_{Y} injective, Hom(X, Y) is injective.

Let Z be another variety, p_{XY}, p_{Y Z}, p_{XZ} be the projections from X × Y × Z to X × Y, Y × Z and X × Z, respectively; p_{X}, p_{Y} be the projections from X × Y to X, Y; p′_{X}, p_{Z} be the projections from X × Z to X, Z, and p′_{Y}, p′_{Z} be the projections from Y × Z to Y and Z.

We define a map in C(X × Y × Z, R)

ψ_{XY Z} : p_{XY}^{∗} Hom(X, Y) ⊗ p_{Y Z}^{∗} Hom(Y, Z) → p_{XZ}^{!} Hom(X, Z)[−2 dim Y]

as the composition of the following (up to twist and shift):

p_{XY}^{∗} Hom(p_{X}^{∗}C_{X}, p_{Y}^{!} C_{Y}) ⊗ p_{Y Z}^{∗} Hom(p′_{Y}^{∗} C_{Y} , p′_{Z}^{!}C_{Z})
→ Hom(p_{XY}^{∗} p_{X}^{∗}C_{X}, p_{XY}^{∗} p_{Y}^{!} C_{Y}) ⊗ Hom(p_{Y Z}^{!}p′_{Y}^{∗} C_{Y} , p_{Y Z}^{!}p′_{Z}^{!}C_{Z})  (1)
→ Hom(p_{XY}^{∗} p_{X}^{∗}C_{X}, p_{Y Z}^{!}p′_{Z}^{!}C_{Z})  (2)
= Hom(p_{XZ}^{∗}p′_{X}^{∗}C_{X}, p_{XZ}^{!}p_{Z}^{!}C_{Z})  (3)
=p_{XZ}^{!} Hom(p′_{X}^{∗}C_{X}, p_{Z}^{!}C_{Z}) .  (4)

We have used (1) The canonical maps

f^{∗} Hom(A, B) → Hom(f^{∗}A, f^{∗}B), f^{∗} Hom(A, B) → Hom(f^{!}A, f^{!}B)

induced by the functoriality of f^{∗} and f^{!} respectively.

(2) The canonical map p_{XY}^{∗} p_{Y}^{!} A → p_{Y Z}^{!}p′_{Y}^{∗} A in (5.2), (2).

(3) The isomorphism p_{Y Z}^{!}p′_{Z}^{!} = p_{Y Z}^{!}p′_{X}^{!} in (5.2), (3).

(4) The formula f^{!} Hom(A, B) = Hom(f^{∗}A, f^{!}B) (see [Bo, p.161]).

(5.3) Proposition. (1) The following diagram commutes in D^{b}(X × Y × Z, R):

p_{XY}^{∗} Hom(X, Y) ⊗ p_{Y Z}^{∗} Hom(Y, Z)  ⟶^{ψ} p_{XZ}^{!} Hom(X, Z)[−2 dim Y]
↑ p_{XY}^{∗} α_{XY} ⊗ p_{Y Z}^{∗}α_{Y Z}  ↑ γ
p_{XY}^{∗} R_{X×Y} ⊗ p_{Y Z}^{∗}R_{Y ×Z}  p_{XZ}^{∗} Hom(X, Z)
∥  ↑ p_{XZ}^{∗}α_{XZ}
R_{X×Y ×Z} ⊗ R_{X×Y ×Z}  ⟶  R_{X×Y ×Z}

where the bottom horizontal arrow is the multiplication, and γ = γ_{p_{XZ}}, the canonical isomorphism in (5.1).

(2) The map ψ is associative in the following sense. Let W be a fourth variety, and π_{XY}, π_{Y Z}, etc. be the projections from X × Y × Z × W to X × Y, Y × Z, etc. The two “composition” maps

π_{XY}^{∗} Hom(X, Y) ⊗ π_{Y Z}^{∗} Hom(Y, Z) ⊗ π_{ZW}^{∗} Hom(Z, W)
→ π_{XW}^{!} Hom(X, W)[−2 dim Y − 2 dim Z]

coincide. One of them is defined using ψ_{XY Z} and ψ_{XZW}, and the other using ψ_{Y ZW} and ψ_{XY W}.

Proof. (1) The isomorphism α_{XY} is concretely described as follows. Let a ∈ R_{X×Y} (U) be a section on U. Then one has a map β∘a : R_{X×Y} → R_{X×Y} → p_{Y}^{!} C_{Y} [−2 dim X]. Since the target is injective, one has a map ã : p_{X}^{∗}C_{X} → p_{Y}^{!} C_{Y} [−2 dim X] on U such that the following square commutes (over U):

p_{X}^{∗}C_{X} ⟶^{ã} p_{Y}^{!} C_{Y} [−2 dim X]
ρ↑  ^{β}↑
R_{X×Y} ⟶^{a}  R_{X×Y}

Under the α_{XY}, a corresponds to ã. The claim (1) is proven using this description and the definition of ψ.

(2) Let (X_{1}, X_{2}, X_{3}, X_{4}) = (X, Y, Z, W). For i ∈ {1, 2, 3, 4}, let {j, k, ℓ} (j < k < ℓ) be the complement of i, and

p_{X_{i}} : X_{i} × X_{j} → X_{i} ,  p′_{X_{i}} : X_{i} × X_{k} → X_{i} ,  p′′_{X_{i}} : X_{i} × X_{ℓ} → X_{i}

be the projections. For {i, j} ⊂ {1, 2, 3, 4}, let {k.ℓ} (k < ℓ) be its complement, and

p_{X_{i}X_{j}} : X_{i} × X_{j} × X_{k} → X_{i} × X_{j} , p′_{X_{i}X_{j}} : X_{i} × X_{j} × X_{ℓ} → X_{i} × X_{j}

be the projections.

One of the composition maps is defined as follows.

π_{XY}^{∗} Hom(X, Y) ⊗ π_{Y Z}^{∗} Hom(Y, Z) ⊗ π_{ZW}^{∗} Hom(Z, W)
=p_{XY Z}^{∗}p_{XY}^{∗} Hom(X, Y) ⊗ p_{XY Z}^{∗}p_{Y Z}^{∗} Hom(Y, Z) ⊗ p_{XZW}^{∗} p_{ZW}^{∗} Hom(Z, W)
⟶p_{XY Z}^{∗}p_{XZ}^{!} Hom(X, Z) ⊗ p_{XZW}^{∗} p_{ZW}^{∗} Hom(Z, W)[−2 dim Y]  (1)
⟶p_{XZW}^{!} p′_{XZ}^{∗} Hom(X, Z) ⊗ p_{XZW}^{∗} p_{ZW}^{∗} Hom(Z, W)[−2 dim Y]  (2)
⟶p_{XZW}^{!} (p′_{XZ}^{∗} Hom(X, Z) ⊗ p_{ZW}^{∗} Hom(Z, W))[−2 dim Y]  (3)
⟶p_{XZW}^{!} p_{XW}^{!} Hom(X, W)[−2 dim Y − 2 dim Z]  (4)
=π_{XW}^{!} Hom(X, W)[−2 dim Y − 2 dim Z] .

(1) is the map p_{XY Z}^{∗}ψ_{XY Z} ⊗1. (2) is ϵ⊗1. (3) is the canonical map f^{!}A⊗f^{∗}B → f^{!}(A⊗B) induced from f^{∗}B → f^{∗} Hom(A, A ⊗ B) → Hom(f^{!}A, f^{!}(A ⊗ B)). (4) is p_{XZW}^{!} ψ_{XZW}.

This map can be concretely described as follows. Let u ∈ π_{XY}^{∗} Hom(p_{X}^{∗}C_{X}, p_{Y}^{!} C_{Y}), v ∈ π_{Y Z}^{∗} Hom(p′_{Y}^{∗} C_{X}, p′_{Z}^{!}C_{Z}), and w ∈ π_{ZW}^{∗} Hom(p′′_{Z}^{∗}C_{Z}, p′′_{W}^{!} C_{W}). They give rise to maps

u : p_{XY Z}^{∗}p_{XY}^{∗} p_{X}^{∗}C_{X} → p_{XY Z}^{∗}p_{XY}^{∗} p_{Y}^{!} C_{Y} ,
v : p_{XY Z}^{∗}p_{Y Z}^{!}p′_{Y}^{∗} C_{Y} → p_{XY Z}^{∗}p_{Y Z}^{!}p′_{Z}^{!}C_{Z} ,
w : p_{XZW}^{!} p_{ZW}^{!} p′′_{Z}^{∗}C_{Z} → p_{XZW}^{!} p_{ZW}^{!} p′′_{W}^{!} C_{W} .

The composition map defined above sends u ⊗ v ⊗ w to a section of Hom(p_{XY Z}^{∗}p_{XY}^{∗} p_{X}^{∗}C_{X}, p_{XZW}^{!} p_{ZW}^{!} p′′_{W}^{!} C_{W}) given as

p_{XY Z}^{∗}p_{XY}^{∗} p_{X}^{∗}C_{X} ⟶^{u}p_{XY Z}^{∗}p_{XY}^{∗} p_{Y}^{!} C_{Y}
⟶^{p_{XY Z}^{∗}(ϵ)}p_{XY Z}^{∗}p_{Y Z}^{!}p′_{Y}^{∗} C_{Y}
⟶^{v}p_{XY Z}^{∗}p_{Y Z}^{!}p′_{Z}^{!}C_{Z} = p_{XY Z}^{∗}p_{XZ}^{!}p_{Z}^{!}C_{Z}
⟶^{ϵ}p_{XZW}^{!} p′_{XZ}^{∗}p_{Z}^{!}C_{Z}
⟶^{p_{XZW}^{!} (ϵ)}p_{XZW}^{!} p_{ZW}^{!} p′′_{Z}^{∗}C_{Z}
⟶^{w}p_{XZW}^{!} p_{ZW}^{!} p′′_{W}^{!} C_{Z} .

The other composition map is defined in a similar manner, and concretely described exactly the same way. So the two maps coincide.

Let X be a smooth projective variety, and D a variety. Let f : X × D → X and g : X × D → D be the two projections. For a sheaf (or a complex of sheaves) A on X, one considers

C_{D}(A) := C(g_{∗}f^{∗}A)

the Godement resolution of the sheaf g_{∗}f^{∗}A on D. For a closed set V ⊂ X × D one also has (g_{V ∗} is the direct image with support)

C_{V,D}(A) := C(g_{V ∗}f^{∗}A) .

(5.4) Proposition. (1) Let A be a complex of sheaves on X. To a map α : D′ → D there is associated a canonical map

α^{∗} : α^{∗}C_{D}(A) → C_{D′}(A) .

Given V ⊂ X × D the above map restricts to a map

α^{∗} : α^{∗}C_{V,D}(A) → C_{(1×α)^{−1}V,D′}(A) .

(2) For a proper map h : X → Y and a complex of sheaves A on X (and V ⊂ X × D) there is a canonical isomorphism

h_{∗} : C_{D}(A) → C_{D}(h_{∗}A)

and a canonical map

h_{∗} : C_{V,D}(A) → C_{(h×1_{D})(V)}(h_{∗}A) .

(3) For a map h : X → Y and a complex of sheaves B on Y (and W ⊂ Y × D) there are canonical maps

h^{∗} : C_{D}(B) → C_{D}(h^{∗}B) ,
h^{∗} : C_{W,D}(B) → C_{(h×1_{D})^{−1}W,D}(h^{∗}B) .

(4) If A and B are on X (and V, W ⊂ X × D) there are canonical maps

C_{D}(A) ⊗ C_{D}(B) → C_{D}(A ⊗ B) ,
C_{V,D}(A) ⊗ C_{W,D}(B) → C_{V ∩W,D}(A ⊗ B) .

Proof. (1) Let the maps be as labeled:

X ⟵^{f′}  X × D′  ⟶^{g′} D′
∥  ↓ 1 × α  ↓ α
X ⟵^{f}  X × D  ⟶^{g} D .

The map in question is the natural one which makes the following diagram commutative

α^{∗}g_{∗}f^{∗}A →^{∼}  g′_{∗}f′^{∗}A
↓  ↓
α^{∗}C_{D}(A) ⟶  C_{D}(A′) .

Similarly the map α^{∗}g_{V ∗}f^{∗}A → (g′_{(1×α)^{−1}V})_{∗}f′^{∗}A induces the second map.

(2) Let the maps be as in:

X  ⟶^{h}  Y
↑ f  ↑ f′
X × D ⟶^{h×1} Y × D
↓ g  ↓ g′
D  =  D

One has g_{∗}f^{∗}A = g′_{∗}f′^{∗}h_{∗}A, whence the isomorphism C_{D}(A) → C_{D}(h_{∗}A).

(3) With the same notation as in (2), there is a canonical map g′^{∗}f′^{∗}B → g_{∗}f^{∗}h^{∗}B, and the map C_{D}(B) → C_{D}(h^{∗}B) is induced from it.

(4) The map is induced from the natural map g_{∗}f^{∗}A ⊗ g_{∗}f^{∗}B → g_{∗}f^{∗}(A ⊗ B).

(5.5) Proposition. Let R_{X} → A be an injective resolution. Then

H^{p}(X × D, R_{X×D}) = H^{p}(D, C_{D}(A)) = H^{p}Γ(D, C_{D}(A))

and

H_{V}^{p} (X × D, R_{X×D}) = H^{p}(D, C_{V,D}(A)) = H^{p}Γ(D, C_{V,D}(A)) .

Proof. One has maps in D(D, R)

C_{D}(A) ← g_{∗}f^{∗}A → ℝg_{∗}f^{∗}A .

The left map is a quasi-isomorphism. The right one is also a quasi-isomorphism since f^{∗}A is fiberwise injective for the map g, hence g_{∗}-acyclic. Hence

C_{D}(A) ≅ ℝg_{∗}f^{∗}A ≅ ℝg_{∗}R_{X×D} .

Since C_{D}(A) is injective, one has the claim. For supported cohomology one has C_{V,D}(A) ≅ g_{V ∗}f^{∗}A ≅ ℝg_{V ∗}f^{∗}A ≅ ℝg_{V ∗}R_{X×D}.

Applying the above to Hom(X, Y) on X × Y and a variety D, we have C_{D}(Hom(X, Y)), a complex of sheaves on D. Let

Hom(X, Y)_{D} := Γ(D, C_{D}(Hom(X, Y))) .

Similarly for V ⊂ X×Y ×D one has a complex of sheaves of R-modules C_{V,D}(Hom(X, Y)) on D, and a complex of R-modules Hom(X, Y)_{V,D} := Γ(D, C_{V,D}(Hom(X, Y))). These objects have the properties as stated below.

(5.6) Properties of C_{D}(Hom(X, Y)) and Hom(X, Y)_{D}. (1) The complex Hom(X, Y)_{D} has p-th cohomology equal to H^{p}(X×Y ×D, R); the Hom(X, Y)_{V,D} has cohomology H_{V}^{p} (X× Y × D, R).

(2) One has the map

ψ_{XY Z} : C_{D}(Hom(X, Y)) ⊗ C_{D}(Hom(Y, Z)) → C_{D}(Hom(X, Z))[−2 dim Y]

defined as the composition

C_{D}(Hom(X, Y)) ⊗ C_{D}(Hom(Y, Z))

→ C_{D}(p_{XY} ^{∗} Hom(X, Y)) ⊗ C_{D}(p_{Y Z}^{∗} Hom(Y, Z))  by (5.4), (3)
→ C_{D}(p_{XY} ^{∗} Hom(X, Y) ⊗ p_{Y Z}^{∗} Hom(Y, Z))  by (5.4), (4)
→ C_{D}(p_{XZ}^{!} Hom(X, Z)[−2 dim Y])  by the map ψ

→ C_{D}(Hom(X, Z)[−2 dim Y]) by the trace map and (5.4), (2).

The map is associative in the obvious sense. The ψ induces the map of complexes

ψ_{XY Z} : Hom(X, Y)_{D} ⊗ Hom(Y, Z)_{D} → Hom(X, Z)_{D}[−2 dim Y] .

If V ⊂ X × Y × D, W ⊂ Y × Z × D are closed subsets, one has

ψ : C_{V,D}(Hom(X, Y)) ⊗ C_{W,D}(Hom(Y, Z)) → C_{V ×Z∩X×W} (Hom(X, Z))[−2 dim Y]

and ψ : Hom(X, Y)_{V,D} ⊗ Hom(Y, Z)_{W,D} → Hom(X, Z)_{V ×Z∩X×W} [−2 dim Y] .

(3) To a map α : D′ → D there are associated pull-backs

α^{∗}C_{D}(Hom(X, Y)) → C_{D′}(Hom(X, Y))

and

α^{∗} : Hom(X, Y)_{D} → Hom(X, Y)_{D′} .

(4) Let D′ be another variety. If V ⊂ X × Y × D and W ⊂ Y × Z × D′ are given we have “composition” maps, satisfying associativity

Hom(X, Y)_{D} ⊗ Hom(Y, Z)_{D′} → Hom(X, Z)_{D×D′}[−2 dim Y]

and

Hom(X, Y)_{V,D} ⊗ Hom(Y, Z)_{W,D′} → Hom(X, Z)_{W ∘V,D×D′}[−2 dim Y] .

To define the former map, one uses the pull-back maps Hom(X, Y)_{D} → Hom(X, Y)_{D×D′} and Hom(Y, Z)_{D′} → Hom(Y, Z)_{D×D′}, and the composition map (2) over D × D′.

Now we take as D the cubical scheme □^{•}. For each b ≤ 0 one defines

C^{• b}\underline{Hom}((X, r), (Y, s)) := C_{□^{−b}}(Hom(X, Y))[2s − 2r] ,

a complex of injective sheaves on □^{−b}, and in particular

C^{• b}(X, r) := C^{• b}\underline{Hom}((pt, 0), (X, r)) = C_{□^{−b}}(C_{X})[2r] .

The complex Γ(□^{−b}, C^{• b}\underline{Hom}((X, r), (Y, s))), which is Hom(X, Y)_{□^{−b}}[2s − 2r] in the previous notation, has cohomology H^{∗+2s−2r}(X × Y × □^{−b}, R).

For simplicity of notation set

ΓC^{• b}\underline{Hom}((X, r), (Y, s)) := Γ(□^{−b}, C^{• b}\underline{Hom}((X, r), (Y, s)))^{alt},
ΓC^{• b}(X, r) := Γ(□^{−b}, C^{• b}(X, r))^{alt} .

Here alt means alternating part with respect to the action of symmetric group Σ_{−b} on □^{−b}. Define

∂ : ΓC^{• b}\underline{Hom}((X, r), (Y, s)) → ΓC^{• b+1}\underline{Hom}((X, r), (Y, s))

to be the alternating sum of the maps induced by the face maps □^{−b−1} → □^{−b}. One has a double complex ΓC^{••}\underline{Hom}((X, r), (Y, s)) with differentials d (of each ΓC^{• b}) and ∂. Its associated simple complex is denoted by ΓC\underline{Hom}((X, r), (Y, s)) without superscripts; it has differential D = d + (−1)^{b}∂. Similarly ΓC(X, r) denotes the simple complex of ΓC^{••}(X, r).

For each b fixed, the complex ΓC^{• b}C\underline{Hom}((X, r), (Y, s)) has pth cohomology equal to H^{p+2s−2r}(X×Y ×□^{−b}, R)^{alt}, which is zero if b < 0. Thus the complex ΓC\underline{Hom}((X, r), (Y, s)) has pth cohomology H^{p+2s−2r}(X × Y, R). Similarly ΓC(X, r) has pth cohomology equal to H^{p+2r}(X, R).

Given Z ∈ Z^{r}(X, n) define a subcomplex

\begin{aligned}Γ_{|Z|}C^{••}(X, r) & ⊂ ΓC^{•,≥−n}(X, r) \\ & ⊂ ΓC^{••}(X, r) \end{aligned}

where

Γ_{|Z|}C^{a,b}(X, r) = Γ_{∪_{δ}|δ^{∗}Z|}(□^{−b}, C^{a,b}(X, r))^{alt}

(δ varies over the face maps δ : □^{−b} → □^{n}). Similarly for f ∈ Z^{0}(\underline{Hom}((X, r), (Y, s)))^{−n} one can define a subcomplex

Γ_{|f|}C^{••}\underline{Hom}((X, r), (Y, s)) ⊂ ΓC^{••}\underline{Hom}((X, r), (Y, s)) .

By (5.6), (4) there is a map of complexes

ΓC^{••}\underline{Hom}((X, r), (Y, s)) ⊗ ΓC^{•,•}\underline{Hom}((Y, s), (Z, t)) → ΓC^{••}\underline{Hom}((X, r), (Z, t)) ,
u ⊗ v  ↦v∘u ,

and u ⊗ v → v∘u is associative. In particular, there is a map

ΓC^{••}(X, r) ⊗ ΓC^{••}\underline{Hom}((X, r), (Y, s)) → ΓC^{••}(Y, s) , α ⊗ u ↦ u_{∗}(α).

We would like to define similar operations for supported subcomplexes. Suppose given smooth projective varieties X, Y, Z and cycles

f ∈ Z^{0}(\underline{Hom}( (X, r), (Y, s)))^{−a}, g ∈ Z^{0}(\underline{Hom}( (Y, s), (Z, t)))^{−b} .

We have already defined

Γ_{|f|}C^{••}\underline{Hom}((X, r), (Y, s)) and Γ_{|g|}C^{••}\underline{Hom}((Y, s), (Z, t)) .

Define now the subcomplex

Γ_{|g∘f|}C^{••}\underline{Hom}((X, r), (Z, t)) ⊂ ΓC^{••}\underline{Hom}((X, r), (Z, t))

by the support condition (let ℓ = dim X + a, m = dim Y + b so f is a cycle on X × Y × □^{ℓ} and g on Y × Z × □^{m})

⋃_{δ:□^{−b}→□^{ℓ+m}} |δ^{∗}p_{XZ ∗}(f × Z × □^{m} · X × g × □^{ℓ})| .

Note the imposed condition is stronger than the one given by the cycle g∘f; recall the g∘f is the alternation of (f × Z × □^{m}) · (X × g × □^{ℓ}). One then has a map

Γ_{|f|}C^{••}\underline{Hom}((X, r), (Y, s)) ⊗ Γ_{|g|}C^{••}\underline{Hom}((Y, s), (Z, t)) → Γ_{|g∘f|}C^{••}\underline{Hom}((X, r), (Z, t)) .

More generally given a finite sequence of varieties X_{0}, X_{1}, ⋯, X_{N} and cycles f^{i,i+1} ∈ Z^{0}(\underline{Hom}((X_{i}, r_{i}), (X_{i+i}, r_{i+1})))^{−a_{i}} such that f^{N−1,N} ∘ ⋯ ∘f^{0,1} is defined, one can define a subcomplex

Γ_{|f^{N−1,N} ∘···∘f^{0,1}|}C^{••}\underline{Hom}((X_{0}, r_{0}), (X_{N}, r_{N}))

which is the target of the composition map from

Γ_{|f^{0,1}|}C^{••}\underline{Hom}((X_{0}, r_{0}), (X_{1}, r_{1})) ⊗ ⋯ ⊗ Γ_{|f^{N−1,N} |}C^{••}\underline{Hom}((X_{N−1}, r_{N−1}), (X_{N}, r_{N})) .

One can extend the above constructions by linearity to formal symbols. If K = ⊕(X_{α}, r_{α}) and L = ⊕(Y_{β}, s_{β}) let ΓC^{••}(K) = ⊕ΓC^{••}(X_{α}, r_{α}) and

ΓC^{••}(\underline{Hom}(K, L)) = ∏_{α} ⨁_{β} ΓC^{••}\underline{Hom}((X_{α}, r_{α}), (Y_{β}, s_{β})) .

The following is obvious from the discussion above.

Properties of ΓC\underline{Hom}(K, L) and ΓC(K). Let K, L, M be objects in D_{inf}(k). One has:

(1) The composition map

ΓC\underline{Hom}(K, L) ⊗ ΓC\underline{Hom}(L, M) → ΓC\underline{Hom}(K, M), u ⊗ v ↦ v∘u

satisfying associativity.

(2) The map ΓC(K) ⊗ ΓC\underline{Hom}(K, L) → ΓC(L), α ⊗ u ↦ u_{∗}α.

(3) For f ∈ Hom(K, L)^{−n} the supported subcomplex Γ_{|f|}C\underline{Hom}(K, L) of ΓC\underline{Hom}(K, L), and for f ∈ Z^{0}(K, n), the subcomplex Γ_{|f|}C(K) of ΓC(K).

(4) The restriction of the map (1)

Γ_{|f|}C\underline{Hom}(K, L) ⊗ Γ_{|g|}C\underline{Hom}(L, M) → Γ_{|g∘f|}C\underline{Hom}(K, M)

if g∘f is defined.

(5.7) Proposition. (1) Let f ∈ Hom(K, L)^{−n}. For each b the complex

Γ_{|f|}C^{•,b}\underline{Hom}(K, L)

is acyclic in degrees < 0.

(2) The map of complexes induced by the augmentation

ΓC^{•,0}\underline{Hom}(K, L) → ΓC^{••}\underline{Hom}(K, L)

is a quasi-isomorphism.

Proof. (1) One is reduced to the case of Γ_{|f|}C^{•,b}\underline{Hom}((X, r), (Y, s)), and then uses the purity property, see [Mi].

(2) One is reduced to the case ΓC^{•,•}\underline{Hom}((X, r), (Y, s)), which was discussed above.

We recall facts on cycle classes. If f is an algebraic cycle of codimension r, there is associated its cycle class

cl(f) ∈ H_{|f|}^{2r}(X, R)

as well as its image cl(f) ∈ H^{2r}(X, R). The cycle class is functorial: If φ : X′ → X is map of smooth projective varieties, and f is a cycle on X, one has cl(φ^{∗}f) = φ^{∗}cl(f).

The idea for constructing the cohomology realization is the following. Take a diagram K = (K^{m}, f^{m,n}), and consider the complexes

ΓC(K^{m})

for all m. On has cycle classes cl(f^{m,n}) ∈ H^{−(n−m−1)}ΓC^{0,•}\underline{Hom}(K^{m}, K^{n}), so one may take their representatives ∈ ΓC^{0,−(n−m−1)}\underline{Hom}(K^{m}, K^{n}). We will extend these to elements in the double complex F^{m,n} ∈ ΓC^{−(n−m−1)}\underline{Hom}(K^{m}, K^{n}). Then one has induced maps of graded groups

F_{∗}^{m,n} : ΓC(K^{m}) → ΓC(K^{n})[−(n − m − 1)] .

If {F^{m,n}} may be so chosen that (ΓC(K^{m}), F_{∗}^{m,n}) is a C-complex then one may take its total complex. We can achieve this by taking elements F^{m,n} ∈ Γ_{|f|}C^{−(n−m−1)}\underline{Hom}(K^{m}, K^{n}) satisfying a condition similar to the one for {f^{m,n}}.

One may verify the following lemma, which will be used in creating cochains with prescribed coboundaries in a double complex.

(5.8) Lemma. Let (A^{••}; d, ∂) be a double complex, with differentials d, ∂ of degree (1, 0), (0, 1), respectively; let D = d+(−1)^{b} ∂ on A^{a,b}. Suppose for each b ∈ ℤ the complex (A^{•,b}, d) is acyclic in degrees < 0. Also, there is a large enough integer N such that A^{a,b} = 0 for b ≥ N.

(1) Suppose given e = (e^{a,−a+1}) ∈ ⊕_{a≤0}A^{a,−a+1} and c^{0,0} ∈ A^{0,0} such that:

D(e) = 0, d(c^{0,0}) = 0, and [e^{0,1}]_{d} = [∂(c^{0,0})]_{d} in H^{0}(A^{•,1}) .

Then there exists c = (c^{a,−a}) ∈ ⊕_{a≤0}A^{a,−a} having the given c^{0,0} as a component and D(c) = e.

(2) If c = (c^{a,−a}) ∈ ⊕_{a≤0}A^{a,−a} satisfies D(c) = 0 and [c^{0,0}]_{d} = 0, there is f = (f^{a,−a−1}) ∈ ⊕_{a≤−1}A^{a,−a−1} such that D(f) = c.

If c = (c^{a,−a}) ∈ ⊕_{a≤−1}A^{a,−a} satisfies D(c) = 0, there is f = (f^{a,−a−1}) ∈ ⊕_{a≤−2}A^{a,−a−1} such that D(f) = c.

(5.9) Proposition. (1) Let K = (K^{•}; f^{••}) be an object of D_{inf}(k). Then there exist, for m < n, elements

F^{m,n} ∈ ⨁_{a+b=−(n−m−1),a≤0} Γ_{|f|}C^{a,b}(\underline{Hom}(K^{m}, K^{n}))

such that its (0, −n + m + 1)-component ^{0}F^{m,n} ∈ Γ_{|f|}C^{0,−(n−m−1)}\underline{Hom}(K^{m}, K^{n}) satisfies

[^{0}F^{m,n}]_{d} = cl(f^{m,n}) ∈ H^{0}Γ_{|f|}C^{0,−(n−m−1)}(\underline{Hom}(K^{m}, K^{n}))

and one has the relation

(∗)  (−1)^{n}D(F^{m,n}) + ∑_{m<ℓ<n} F^{ℓ,n}∘F^{m,ℓ} = 0

in ⨁_{a+b=−(n−m−2),a≤0} Γ_{|f|}C^{a,b}(\underline{Hom}(K^{m}, K^{n})) .

We call such (F^{m,n}) a representative of (cl(f^{m,n})).

(2) If (F̄^{m,n}) is another representative of (cl(f^{m,n})), there exist, for m < n, elements

I^{m,n} ∈ ⨁_{a+b=−(n−m),a≤−1} Γ_{|f|}C^{a,b}(\underline{Hom}(K^{m}, K^{n}) )

such that

\begin{matrix}(−1)^{n}D(I^{m,n})− & (−1)^{m+n}F^{m,n} + (−1)^{m+n}F̄^{m,n} \\ − ∑_{m<ℓ<n}(−1)^{m+ℓ}I^{ℓ,n}∘F^{m,ℓ} + ∑_{m<ℓ<n}(−1)^{ℓ+n}F̄^{ℓ,n}∘I^{m,ℓ} = 0 . \end{matrix}

Moreover, if (′I^{m,n}) also satisfies the same condition as (I^{m,n}), then there exist

J^{m,n} ∈ ⨁_{a+b=−n+m−1,a≤−2} Γ_{|f|}C^{a,b}(\underline{Hom}(K^{m}, K^{n})) (m < n)

such that

I^{m,n} −′I^{m,n} = (−1)^{n}D(J^{m,n}) + ∑(−1)^{ℓ+n}J^{ℓ,n}∘F^{m,ℓ} + ∑(−1)^{ℓ+n}F̄^{ℓ,n}∘J^{m,ℓ} .

Proof. (1) We will abbreviate the double complex Γ_{|f|}C^{••}\underline{Hom}(K^{m}, K^{n}) by Γ_{|f|}C^{••}(m, n). The term Γ_{|f|}C^{a,b}(m, n) is zero for b > 0 and the complex (Γ_{|f|}C^{•,b}(m, n), d) is acyclic for in degree < 0.

We construct F^{m,n} by induction on n − m. If n = m + 1, one has cl(f^{m,m+1}) ∈ H^{0}Γ_{|f|}C^{•,0}(m, m + 1). Take an element ^{0}F^{m,m+1} ∈ Γ_{|f|}C^{0,0}(m, m + 1) which represents this class:

d(^{0}F^{m,m+1}) = 0 and [^{0}F^{m,m+1}]_{d} = cl(f^{m,m+1}) ∈ H^{0}Γ_{|f|}C^{•,0}(m, m + 1) .

Let F^{m,m+1} = (^{0}F^{m,m+1}) ∈ ⊕_{a≤0}Γ_{|f|}C^{a,−a}(m, m + 1), the element having ^{0}F^{m,m+1} as the only component.

Assume n ≥ m+2. The element ∑ F^{ℓ,n}∘F^{m,ℓ} ∈ ⨁_{a+b=−n+m+2,a≤0} Γ_{|f|}C^{a,b}(m, n) has its (0, −n+m+2)-component equal to ∑^{0}F^{ℓ,n}∘^{0}F^{m,ℓ}, and [∑ ^{0}F^{ℓ,n}∘^{0}F^{m,ℓ}]_{d} = ∑ cl(f^{ℓ,n}∘f^{m,ℓ}) = −(−1)^{n}∂cl(f^{m,n}) . Take^{0}F^{m,n} ∈ Γ_{|f|}C^{0,−n+m+1} such that [^{0}F^{m,n}]_{d} = cl(f^{m,n}). We apply the above lemma (1) to extend ^{0}F^{m,n} to F^{m,n} ∈ ⨁_{a+b=−n+m+1,a≤0} Γ_{|f|}C^{a,b}(m, n) such that (−1)^{n}D(F^{m,n}) = − ∑ F^{m,ℓ}∘F^{m,ℓ} .

(2) We construct I^{m,n} by induction on n − m. If n = m + 1, since [^{0}F^{m,m+1}]_{d} = [^{0}F̄^{m,m+1}]_{d}, the lemma (2) implies that there exists I^{m,m+1} ∈ ⊕_{a≤−1}Γ_{|f|}C^{a,−a−1}(m, n) such that (−1)^{m+1}D(I^{m,m+1}) = −F^{m,m+1} + F̄^{m,m+1}. Assume n ≥ m + 2. One verifies, using the induction hypothesis, applying D to

−(−1)^{m+n}F^{m,n} + (−1)^{m+n}F̄^{m,n} − ∑_{m<ℓ<n}(−1)^{m+ℓ}I^{ℓ,n}∘F^{m,ℓ} + ∑_{m<ℓ<n}(−1)^{ℓ+n}F̄^{ℓ,n}∘I^{m,ℓ}

yields zero. (For instance

D(∑ F̄^{ℓ,n}∘I^{m,ℓ}) = ∑ D(F̄^{ℓ,n})∘I^{m,ℓ} + ∑(−1)^{ℓ+n+1}F̄^{ℓ,n}∘D(I^{m,ℓ})

and D(F̄^{ℓ,n}) and D(I^{m,ℓ}) are known by (1) and induction hypothesis. )

By the lemma (2) there exists I^{m,n} such that

−(−1)^{n}D(I^{m,n}) = F̄^{m,n} − F^{m,n} + ∑ I^{ℓ,n}∘F^{m,ℓ} + ∑ F̄^{ℓ,n}∘I^{m,ℓ} .

The elements (J^{m,n}) connecting the (I)’s are constructed similarly by induction using now the latter half of the lemma (2).

Now we construct the realization of an object (K^{m}, f^{m,n}). Take a representative (F^{m,n}) of cl(f^{m,n}) and let F_{∗}^{m,n} : ΓC(K^{m}) → ΓC(K^{n}) be the map of degree −(n − m − 1) induced by F^{m,n}. The condition (*) on (F^{m,n}) in the above proposition implies the condition for (ΓC(K^{m}), F_{∗}^{m,n}) to be a C-complex, namely

(−1)^{n}D∘F_{∗}^{m,n} + F_{∗}^{m,n}∘(−1)^{m}D + ∑ F^{ℓ,n}∘F^{m,ℓ} = 0 .

We verify that the C-complex is independent of the choice of (F^{m,n}). Let (F̄^{m,n}) be another representative and (I^{m,n}) be as in the above proposition (2). The I^{m,n} together with I^{m,m} = id give a map of C-complexes (id, I_{∗}^{m,n}) : (ΓC(K^{m}), F_{∗}^{m,n}) → (ΓC(K^{m}), F̄_{∗}^{m,n}). This is in fact a homotopy equivalence of C-complexes, the inverse given by (id, −I_{∗}^{m,n}). The second part of (2) implies that the isomorphism (id, I_{∗}^{m,n}) is uniquely determined up to homotopy equivalence. So one has a canonical isomorphism

(ΓC(K^{m}), F_{∗}^{m,n}) → (ΓC(K^{m}), F̄_{∗}^{m,n}) in K(C − Ab)

and a canonical isomorphism

T ot (ΓC(K^{m}), F_{∗}^{m,n}) → T ot (ΓC(K^{m}), F̄_{∗}^{m,n}) in K(Ab) .

If (F̃^{m,n}) is a third choice of a representative, then one checks that the three comparison isomorphisms among T ot (ΓC(K^{m}), F_{∗}^{m,n}), T ot (ΓC(K^{m}), F̄_{∗}^{m,n}), and T ot (ΓC(K^{m}), F̃_{∗}^{m,n}) are compatible. We can thus define

ΓC(K) = T ot (ΓC(K^{m}), F_{∗}^{m,n})

for any choice of a representative.

(5.10) Proposition. Let L = (L^{•}; g^{••}) be another object of D_{inf}(k), and G^{m,n} ∈ ⨁_{a+b=−(n−m−1),a≤0} Γ_{|g|}C^{a,b}(\underline{Hom}(L^{m}, L^{n})) a representative of (cl(g^{m,n})). Let u : K → L

be a morphism in D(k). Let u^{m,n} ∈ Z^{0}(\underline{Hom}(K^{m}, L^{n}))^{−(n−m−1)} represent u, and denote by Γ_{|u|}C^{a,b}(\underline{Hom}(K^{m}, L^{n})) the subcomplex with the support condition with respect to all the compositions g∘ ⋯ ∘g∘u∘f ⋯ ∘f.

(1) There exist

U^{m,n} ∈ ⨁_{a+b=−n+m,a≤0} Γ_{|u|}C^{a,b}(\underline{Hom}(K^{m}, L^{n}))

such that its (0, −n + m)-component ^{0}U^{m,n} ∈ Γ_{|u|}C^{0,−(n−m−1)}(\underline{Hom}(K^{m}, L^{n})) satisfies

[^{0}U^{m,n}]_{d} = cl(u^{m,n}) ∈ H^{0}Γ_{|u|}C^{•,−(n−m−1)}(\underline{Hom}(K^{m}, L^{n}))

and one has the relation

(−1)^{n} D(U^{m,n}) − ∑_{m<ℓ≤n}(−1)^{m+ℓ}U^{ℓ,n}∘F^{m,ℓ} + ∑_{m≤ℓ<n} G^{ℓ,n}∘U^{m,ℓ} = 0 .

We call such (U^{m,n}) a representative of cl(u^{m,n}) with respect to (F) and (G). (2) If (Ū^{m,n}) is another representative, there exist

A^{m,n} ∈ ⨁_{a+b=−n+m−1,a≤−1} Γ_{|u|}C^{a,b}(\underline{Hom}(K^{m}, L^{n})) (m ≤ n)

such that

U^{m,n} − Ū^{m,n} = (−1)^{n}D(A^{m,n}) + ∑(−1)^{m+ℓ}A^{ℓ,n}∘G^{m,ℓ} + ∑(−1)^{ℓ+n}G^{ℓ,n}∘A^{m,ℓ} .

Proof. The proof of (1), (2) is similar to that of the previous proposition, using now the double complex Γ_{|u|}C^{••}(\underline{Hom}(K^{m}, L^{n})). A statement on the ambiguity of (A^{m,n}) may be made, but is omitted since it will not be used.

Proof of Theorem (5.0). For a morphism u : K → L with a representative (u^{m,n}), take a representative (U^{m,n}) of (cl(u^{m,n})); then the maps

U_{∗}^{m,n} : (ΓC(K^{m}), F_{∗}^{m,n}) → (ΓC(L^{m}), G_{∗}^{m,n})

form a map of C-complexes. If (Ū^{m,n}) is another representative of the same (cl(u^{m,n})), take a “homotopy” (A^{m,n}) as in (2) above; then the maps of C-complexes (U_{∗}^{m,n}) and (Ū_{∗}^{m,n}) are homotopic, a homotopy being provided by (A_{∗}^{m,n}). Let, then

u_{∗}((F^{m,n}), (G^{m,n})) := tot (U_{∗}^{m,n}) : T ot (ΓC(K^{m}), F_{∗}^{m,n}) → T ot (ΓC(L^{m}), G_{∗}^{m,n})

(well-defined as a map in the homotopy category K(Ab)).

The map u_{∗}(F, G) = u_{∗}((F^{m,n}), (G^{m,n})) is functorial. Precisely speaking, let (M^{•}, h^{••}) be a third object with a representative (H^{m,n}) of (cl(h^{m,n})) and v : L → M be a morphism. Then

(v∘u)_{∗}(F, H) = v_{∗}(G, H)∘u_{∗}(F, G) : T ot (ΓC(K^{m}), F_{∗}^{m,n}) → T ot (ΓC(M^{m}), H_{∗}^{m,n}) .

This follows from: if (U^{m,n}) is a representative of (cl(u^{m,n})) with respect to F, G and (V^{m,n}) a representative of (cl(v^{m,n})) with respect to G, H, then (∑_{ℓ} V^{ℓ,n}∘U^{m,ℓ}) is a representative of (cl (∑_{ℓ} v^{ℓ,n}∘u^{m,ℓ})) with respect to F, H.

We show that u_{∗}(F, G) is depends only on u, i.e. independent of the choice of the representatives (F), (G). If (F̄) is another representative of cl(u), and and (Ḡ) of cl(v), by the above functoriality, the diagram

T ot (ΓC(K^{m}), F_{∗}) ⟶^{u_{∗}(F,G)} T ot (ΓC(L^{m}), G_{∗})
↓^{id_{∗}(F,F̄)}  ↓^{id_{∗}(G,Ḡ)}
T ot (ΓC(K^{m}), F̄_{∗}) ⟶^{u_{∗}(F̄,Ḡ)} T ot (ΓC(L^{m}), Ḡ_{∗})

is commutative in K(Ab).

So define

u_{∗} = u_{∗}(F, G)

for any choice of representatives. One has (v∘u)_{∗} = v_{∗}∘u_{∗}. Therefore we have a functor

ℝΓ : D_{inf}(k) → D(Λ − vect); K ↦ T ot ΓC(K)

compatible with the functor h.

If K′ → K → K′′ ⟶^{[1]} is a standard distinguished triangle in D_{inf}(k), there is induced

a standard distinguished triangle ΓC(K′) → ΓC(K) → ΓC(K′′) ⟶^{[1]} in K(C − Ab) and consequently one in K(Ab). So the ℝΓ is an exact functor.

§6. Cohomological motives of quasi-projective varieties.

In this section we assume that resolutions of singularities exist for varieties over k (e.g. char(k) = 0). More details are in [Ha 2].

Recall the functor

h : (Smooth P roj./k)^{opp} → D_{finite}(k) .  (6-1)

For X in (Smooth P roj./k), h(X) gives both the cycle complex Z^{r}(X, ·) and the cohomology ℝΓ(X, Λ). Here we consider either Betti cohomology (Λ = ℚ) or etale cohomology (Λ = ℚ_{ℓ}), see §5. We can extend the h in two ways. The first one is the contravariant functor (called the cohomological motive functor)

h : (Q − P roj./k)^{opp} → D_{finite}(k) .

where (Q − P roj./k) is the category of quasi-projective (possibly singular) varieties and maps between them. The second one is the contravariant functor (the cohomological motive with compact support functor)

h_{c} : (Q − P roj./k; proper maps)^{opp} → D_{finite}(k) .

where (Q − P roj./k; proper maps) is the category of quasi-projective varieties and proper (in fact projective) maps between them. We state in more detail.

(6.1) Theorem. There is a contravariant functor

h_{c} : (Q − P roj./k; proper maps) → D_{finite}(k)

satisfying the following properties.

(1) It coincides with the functor h in (6-1) on the subcategory of smooth projective varieties.

(2) It is covariant functorial for open immersions: for an open immersion j : U′ → U there is associated a morphism j_{∗} : h_{c}(U′) → h_{c}(U); one has (j∘j′)_{∗} = j_{∗}∘j′_{∗}.

(3) (Localization) If i : Z → U is a closed immersion and j : U − Z → U is the open immersion of its complement, one has a distinguished triangle

h_{c}(U − Z) →^{j_{∗}} h_{c}(U) →^{i^{∗}} h_{c}(Z) ⟶^{[1]}

functorial for pairs (U, Z).

(4) There is a canonical isomorphism in the homotopy category

Z^{−s}(h_{c}(U)^{∨}, ·) = Z_{s}(U, ·) .

(6.2) Theorem. There is a unique contravariant functor

h : (Q − P roj./k) → D_{finite}(k)

satisfying the following properties.

(5) It coincides with the functor h in (6-1) on the subcategory of smooth projective varieties.

(6) The functor h satisfies the Mayer-Vietoris property: Let

Z′ ⟶^{α} U′
↓^{g}  ↓^{f}
Z ⟶^{β} U

be a Cartesian square of quasi-projective varieties such that the horizontal arrows are closed immersions, and f induces U′ − Z′ →^{∼} U − Z. Then there is a distinguished triangle of the form

h(U) ⟶^{(f^{∗},β^{∗})} h(U′) ⊕ h(Z) ⟶^{g^{∗}−α^{∗}} h(Z′) ⟶^{[1]}

functorial for such Cartesian squares.

(7) There is a canonical morphism h_{c}(U) → h(U). If U is projective, it is an isomorphism.

(8) (Poincaré duality) For U smooth and equi-dimensional, there is a canonical isomorphism

h(U) →^{∼} h_{c}(U)^{∨}(− dim U)[−2 dim U] .

Further properties of the functors h and h_{c} are listed below.

(9) There is the relative cohomology motive functor. A pair of varieties is (U, Z) where U is a quasi-projective variety and Z ⊂ U a closed subset. A variety U may be regarded as the pair (U, ∅). A map of of pairs f : (U′, Z′) → (U, Z) is a map f : U′ → U such that f^{−1}(Z) ⊂ Z′. Denote by (P air of Q − P roj./k) the category of pairs of varieties and maps between them. There exists a functor

h : (P air of Q − P roj./k) → D_{finite}(k) ; (U, Z) → h(U, Z)

which extends the functor h above together with distinguished triangles

h(U, Z) → h(U) → h(Z) ⟶^{[1]} .

(10) A strict map of pairs f : (U′, Z′) → (U, Z) is a map f : U′ → U such that f^{−1}Z = Z′. A variety U may be regarded as the pair (U, U). There is a functor

h : (P air of Q − P roj./k; strict maps) → D_{finite}(k) ; (U, Z) → h_{Z}(U)

which extends the functor h (in the sense h_{U}(U) = h(U)) together with functorial distinguished triangles

h_{Z}(U) → h(U) → h(U − Z) ⟶^{[1]} .

(11) (Purity) If U is smooth equi-dimensional, there is a canonical isomorphism

h_{c}(Z)^{∨}(− dim U)[−2 dim U] →^{∼} h_{Z}(U) .

(6.3) Definition. For a quasi-projective variety U, we define its (co)homological cycle complex (with or without compact support) as follows:

Z^{r}(U, ·)^{∗} =Z^{r}(h(U), ·) , cohomological cycle complex;
Z_{cpt}^{r}(U, ·)^{∗} =Z^{r}(h_{c}(U), ·) , compactly supported cohomological cycle complex;
Z_{s}(U, ·)_{∗} =Z^{−s}(h_{c}(U)^{∨}, ·) , homological cycle complex ;
Z_{s}^{cpt}(U, ·)_{∗} =Z^{−s}(h(U)^{∨}, ·) , compactly supported homological cycle complex .

These are defined as objects in D(Ab). Note Z_{s}(U, •)_{∗} = Z_{s}(U, •), the cycle complex of U.

By definition, the higher Chow cohomology/homology groups (with compact support) are given by:

CHC^{r}(U, n) =H^{−n}Z^{r}(U, ·)^{∗} , Chow cohomology;
CHC_{cpt}^{r}(U, n) =H^{−n}Z_{cpt}^{−s}(U, ·)^{∗} , Chow cohomology with compact support;
CH_{s}(U, n) =H^{−n}Z_{s}(U, ·)_{∗} , Chow homology;
CH_{s}^{cpt}(U, n) =H^{−n}Z_{s}^{cpt}(U, ·)_{∗} , Chow homology with compact support .

Note the Chow homology is nothing but Bloch’s higher Chow group.

Let K ↦ H^{i}(K, Λ) be one of the cohomology realizations in §5. The functors h and h_{c} give the expected cohomology:

(6.4) Theorem. For a quasi-projective variety U, H^{i}(h(U); Λ) = H^{i}(U, Λ), i.e. h(U) calculates the cohomology of U. Similarly,

H^{i}(h_{c}(U); Λ) = H_{cpt}^{i}(U, Λ) ;
H^{i}(h_{c}(U)^{∨}; Λ) = H_{−i}^{BM}(U, Λ) , Borel-Moore homology ;
H^{i}(h(U)^{∨}; Λ) = H_{−i}^{cpt}(U, Λ) , homology (with compact support) .

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Graduate School of Mathematics Kyushu University Fukuoka, 812-8581 Japan