Lemma 45.7.1. Assume given (D1) and (D3) satisfying (A). For $f : X \to Y$ a morphism of smooth projective varieties we have $f_*(f^*b \cup a) = b \cup f_*a$. If $g : Y \to Z$ is a second morphism of smooth projective varieties, then $g_* \circ f_* = (g \circ f)_*$.

## 45.7 Classical Weil cohomology theories

In this section we define what we will call a classical Weil cohomology theory. This is exactly what is called a Weil cohomology theory in [Section 1.2, Kleiman-cycles].

We fix an algebrically closed field $k$ (the base field). In this section *variety* will mean a variety over $k$, see Varieties, Section 33.3. We fix a field $F$ of characteristic $0$ (the coefficient field). A Weil cohomology theory is given by data (D1), (D2), and (D3) subject to axioms (A), (B), and (C).

The data is given by:

A contravariant functor $H^*$ from the category of smooth projective varieties to the category of graded commutative $F$-algebras.

For every smooth projective variety $X$ a group homomorphism $\gamma : \mathop{\mathrm{CH}}\nolimits ^ i(X) \to H^{2i}(X)$.

For every smooth projective variety $X$ of dimension $d$ a map $\int _ X : H^{2d}(X) \to F$.

We make some remarks to explain what this means and to introduce some terminology associated with this.

Remarks on (D1). Given a smooth projective variety $X$ we say that $H^*(X)$ is the *cohomology* of $X$. Given a morphism $f : X \to Y$ of smooth projective varieties we denote $f^* : H^*(Y) \to H^*(X)$ the map $H^*(f)$ and we call it the *pullback map*.

Remarks on (D2). The map $\gamma $ is called the *cycle class map*. We say that $\gamma (\alpha )$ is the *cohomology class* of $\alpha $. If $Z \subset Y \subset X$ are closed subschemes with $Y$ and $X$ smooth projective varieties and $Z$ integral, then $[Z]$ could mean the class of the cycle $[Z]$ in $\mathop{\mathrm{CH}}\nolimits ^*(Y)$ or in $\mathop{\mathrm{CH}}\nolimits ^*(X)$. In this case the notation $\gamma ([Z])$ is ambiguous and the intended meaning has to be deduced from context.

Remarks on (D3). The map $\int _ X$ is sometimes called the *trace map* and is sometimes denoted $\text{Tr}_ X$.

The first axiom is often called *Poincaré duality*

Let $X$ be a smooth projective variety of dimension $d$. Then

$\dim _ F H^ i(X) < \infty $ for all $i$,

$H^ i(X) \times H^{2d - i}(X) \rightarrow H^{2d}(X) \rightarrow F$ is a perfect pairing for all $i$ where the final map is the trace map $\int _ X$,

$H^ i(X) = 0$ unless $i \in [0, 2d]$, and

$\int _ X : H^{2d}(X) \to F$ is an isomorphism.

Let $f : X \to Y$ be a morphism of smooth projective varieties with $\dim (X) = d$ and $\dim (Y) = e$. Using Poincaré duality we can define a *pushforward*

as the contragredient of the linear map $f^* : H^ i(Y) \to H^ i(X)$. In a formula, for $a \in H^{2d - i}(X)$, the element $f_*a \in H^{2e - i}(Y)$ is characterized by

for all $b \in H^ i(Y)$.

**Proof.**
The first equality holds because

The second equality holds because

This ends the proof. $\square$

The second axiom says that $H^*$ respects the monoidal structure given by products via the *Künneth formula*

Let $X$ and $Y$ be smooth projective varieties. The map

\[ H^*(X) \otimes _ F H^*(Y) \to H^*(X \times Y),\quad a \otimes b \mapsto \text{pr}_1^*a \cup \text{pr}_2^*b \]is an isomorphism.

The third axiom concerns the cycle class maps

The cycle class maps satisfy the following rules

for a morphism $f : X \to Y$ of smooth projective varieties we have $\gamma (f^!\beta ) = f^*\gamma (\beta )$ for $\beta \in \mathop{\mathrm{CH}}\nolimits ^*(Y)$,

for a morphism $f : X \to Y$ of smooth projective varieties we have $\gamma (f_*\alpha ) = f_*\gamma (\alpha )$ for $\alpha \in \mathop{\mathrm{CH}}\nolimits ^*(X)$,

for any smooth projective variety $X$ we have $\gamma (\alpha \cdot \beta ) = \gamma (\alpha ) \cup \gamma (\beta )$ for $\alpha , \beta \in \mathop{\mathrm{CH}}\nolimits ^*(X)$, and

$\int _{\mathop{\mathrm{Spec}}(k)} \gamma ([\mathop{\mathrm{Spec}}(k)]) = 1$.

Remark 45.7.2. Let $X$ be a smooth projective variety. We obtain maps

where the first arrow is as in axiom (B) and $\Delta ^*$ is pullback along the diagonal morphism $\Delta : X \to X \times X$. The composition is the cup product as pullback is an algebra homomorphism and $\text{pr}_ i \circ \Delta = \text{id}$. On the other hand, given cycles $\alpha , \beta $ on $X$ the intersection product is defined by the formula

In other words, $\alpha \cdot \beta $ is the pullback of the exterior product $\alpha \times \beta $ on $X \times X$ by the diagonal. Note also that $\alpha \times \beta = \text{pr}_1^*\alpha \cdot \text{pr}_2^*\beta $ in $\mathop{\mathrm{CH}}\nolimits ^*(X \times X)$ (we omit the proof). Hence, given axiom (C)(a), axiom (C)(c) is equivalent to the statement that $\gamma $ is compatible with exterior product in the sense that $\gamma (\alpha \times \beta )$ is equal to $\text{pr}_1^*\gamma (\alpha ) \cup \text{pr}_2^*\gamma (\beta )$. This is how axiom (C)(c) is formulated in [Kleiman-cycles].

Definition 45.7.3. Let $k$ be an algebraically closed field. Let $F$ be a field of characteristic $0$. A *classical Weil cohomology theory* over $k$ with coefficients in $F$ is given by data (D1), (D2), and (D3) satisfying Poincaré duality, the Künneth formula, and compatibility with cycle classes, more precisely, satisfying (A), (B), and (C).

We do a tiny bit of work.

Lemma 45.7.4. Let $H^*$ be a classical Weil cohomology theory (Definition 45.7.3). Let $X$ be a smooth projective variety of dimension $d$. The diagram

commutes where $\deg : \mathop{\mathrm{CH}}\nolimits _0(X) \to \mathbf{Z}$ is the degree of zero cycles discussed in Chow Homology, Section 42.40.

**Proof.**
The result holds for $\mathop{\mathrm{Spec}}(k)$ by axiom (C)(d). Let $x : \mathop{\mathrm{Spec}}(k) \to X$ be a closed point of $X$. Then we have $\gamma ([x]) = x_*\gamma ([\mathop{\mathrm{Spec}}(k)])$ in $H^{2d}(X)$ by axiom (C)(b). Hence $\int _ X \gamma ([x]) = 1$ by the definition of $x_*$.
$\square$

Lemma 45.7.5. Let $H^*$ be a classical Weil cohomology theory (Definition 45.7.3). Let $X$ and $Y$ be smooth projective varieties. Then $\int _{X \times Y} = \int _ X \otimes \int _ Y$.

**Proof.**
Say $\dim (X) = d$ and $\dim (Y) = e$. By axiom (B) we have $H^{2d + 2e}(X \times Y) = H^{2d}(X) \otimes H^{2e}(Y)$ and by axiom (A)(d) this is $1$-dimensional. By Lemma 45.7.4 this $1$-dimensional vector space generated by the class $\gamma ([x \times y])$ of a closed point $(x, y)$ and $\int _{X \times Y} \gamma ([x \times y]) = 1$. Since $\gamma ([x \times y]) = \gamma ([x]) \otimes \gamma ([y])$ by axioms (C)(a) and (C)(c) and since $\int _ X \gamma ([x]) = 1$ and $\int _ Y \gamma ([y]) = 1$ we conclude.
$\square$

Lemma 45.7.6. Let $H^*$ be a classical Weil cohomology theory (Definition 45.7.3). Let $X$ and $Y$ be smooth projective varieties. Then $\text{pr}_{2, *} : H^*(X \times Y) \to H^*(Y)$ sends $a \otimes b$ to $(\int _ X a) b$.

**Proof.**
This is equivalent to the result of Lemma 45.7.5.
$\square$

Lemma 45.7.7. Let $H^*$ be a classical Weil cohomology theory (Definition 45.7.3). Let $X$ be a smooth projective variety of dimension $d$. Choose a basis $e_{i, j}, j = 1, \ldots , \beta _ i$ of $H^ i(X)$ over $F$. Using Künneth write

with $e'_{2d - i, j} \in H^{2d - i}(X)$. Then $\int _ X e_{i, j} \cup e'_{2d - i, j'} = (-1)^ i\delta _{jj'}$.

**Proof.**
Recall that $\Delta ^* : H^*(X \times X) \to H^*(X)$ is equal to the cup product map $H^*(X) \otimes _ F H^*(X) \to H^*(X)$, see Remark 45.7.2. On the other hand we have $\gamma ([\Delta ]) = \Delta _*\gamma ([X]) = \Delta _*1$ by axiom (C)(b) and the fact that $\gamma ([X]) = 1$. Namely, $[X] \cdot [X] = [X]$ hence by axiom (C)(c) the cohomology class $\gamma ([X])$ is $0$ or $1$ in the $1$-dimensional $F$-algebra $H^0(X)$; here we have also used axioms (A)(d) and (A)(b). But $\gamma ([X])$ cannot be zero as $[X] \cdot [x] = [x]$ for a closed point $x$ of $X$ and we have the nonvanishing of $\gamma ([x])$ by Lemma 45.7.4. Hence

by the definition of $\Delta _*$. On the other hand, we have

by Lemma 45.7.5; note that we made two switches of order so that the sign is $1$. Thus if we choose $a$ such that $\int _ X a \cup e_{i, j} = 1$ and all other pairings equal to zero, then we conclude that $\int _ X e'_{2d - i, j} \cup b = \int _ X a \cup b$ for all $b$, i.e., $e'_{2d - i, j} = a$. This proves the lemma. $\square$

Lemma 45.7.8. Let $H^*$ be a classical Weil cohomology theory (Definition 45.7.3). Let $X$ be a smooth projective variety. We have

**Proof.**
Equality on the right. We have $[\Delta ] \cdot [\Delta ] = \Delta _*(\Delta ^![\Delta ])$ (Chow Homology, Lemma 42.61.6). Since $\Delta _*$ preserves degrees of $0$-cycles it suffices to compute the degree of $\Delta ^![\Delta ]$. The class $\Delta ^![\Delta ]$ is given by capping $[\Delta ]$ with the top Chern class of the normal sheaf of $\Delta \subset X \times X$ (Chow Homology, Lemma 42.53.5). Since the conormal sheaf of $\Delta $ is $\Omega _{X/k}$ (Morphisms, Lemma 29.32.7) we see that the normal sheaf is equal to the tangent sheaf $\mathcal{T}_ X = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\Omega _{X/k}, \mathcal{O}_ X)$ as desired.

Equality on the left. By Lemma 45.7.4 we have

Write $\gamma ([\Delta ]) = \sum e_{i, j} \otimes e'_{2d - i , j}$ as in Lemma 45.7.7. Recalling that $\Delta ^*$ is given by cup product we obtain

as desired. $\square$

We will now tie classical Weil cohomology theories in with motives as follows.

Lemma 45.7.9. Let $k$ be an algebraically closed field. Let $F$ be a field of characteristic $0$. Consider a $\mathbf{Q}$-linear functor

of symmetric monoidal categories such that $G(\mathbf{1}(1))$ is nonzero only in degree $-2$. Then we obtain data (D1), (D2), (D3) satisfying all of (A), (B), (C) except for possibly (A)(c) and (A)(d).

**Proof.**
We obtain a contravariant functor from the category of smooth projective varieties to the category of graded $F$-vector spaces by setting $H^*(X) = G(h(X))$. By assumption we have a canonical isomorphism

compatible with pullbacks. Using pullback along the diagonal $\Delta : X \to X \times X$ we obtain a canonical map

of graded vector spaces compatible with pullbacks. This defines a functorial graded $F$-algebra structure on $H^*(X)$. Since $\Delta $ commutes with the commutativity constraint $h(X) \otimes h(X) \to h(X) \otimes h(X)$ (switching the factors) and since $G$ is a functor of symmetric monoidal categories (so compatible with commutativity constraints), and by our convention in Homology, Example 12.17.4 we conclude that $H^*(X)$ is a graded commutative algebra! Hence we get our datum (D1).

Since $\mathbf{1}(1)$ is invertible in the category of motives we see that $G(\mathbf{1}(1))$ is invertible in the category of graded $F$-vector spaces. Thus $\sum _ i \dim _ F G^ i(\mathbf{1}(1)) = 1$. By assumption we only get something nonzero in degree $-2$ and we may choose an isomorphism $F[2] \to G(\mathbf{1}(1))$ of graded $F$-vector spaces. Here and below $F[n]$ means the graded $F$-vector space which has $F$ in degree $-n$ and zero elsewhere. Using compatibility with tensor products, we find for all $n \in \mathbf{Z}$ an isomorphism $F[2n] \to G(\mathbf{1}(n))$ compatible with tensor products.

Let $X$ be a smooth projective variety. By Lemma 45.3.1 we have

Applying the functor $G$ we obtain

This is the datum (D2).

Let $X$ be a smooth projective variety of dimension $d$. By Lemma 45.3.1 we have

Thus the class of the cycle $[X]$ in $\mathop{\mathrm{CH}}\nolimits _ d(X)$ defines a morphism $h(X)(d) \to \mathbf{1}$. Applying $G$ we obtain

This map is zero except in degree $0$ where we obtain $\int _ X : H^{2d}(X) \to F$. This is the datum (D3).

Let $X$ be a smooth projective variety of dimension $d$. By Lemma 45.4.9 we know that $h(X)(d)$ is a left dual to $h(X)$. Hence $G(h(X)(d)) = H^*(X) \otimes F[-2d]$ is a left dual to $H^*(X)$ in the category of graded $F$-vector spaces. By Homology, Lemma 12.17.5 we find that $\sum _ i \dim _ F H^ i(X) < \infty $ and that $\epsilon : h(X)(d) \otimes h(X) \to \mathbf{1}$ produces nondegenerate pairings $H^{2d - i}(X) \otimes _ F H^ i(X) \to F$. In the proof of Lemma 45.4.9 we have seen that $\epsilon $ is given by $[\Delta ]$ via the identifications

Thus $\epsilon $ is the composition of $[X] : h(X)(d) \to \mathbf{1}$ and $h(\Delta )(d) : h(X)(d) \otimes h(X) \to h(X)(d)$. It follows that the pairings above are given by cup product followed by $\int _ X$. This proves axiom (A) parts (a) and (b).

Axiom (B) follows from the assumption that $G$ is compatible with tensor structures and our construction of the cup product above.

Axiom (C). Our construction of $\gamma $ takes a cycle $\alpha $ on $X$, interprets it as a correspondence $a$ from $\mathop{\mathrm{Spec}}(k)$ to $X$ of some degree, and then applies $G$. If $f : Y \to X$ is a morphism of smooth projective varieties, then $f^!\alpha $ is the pushforward (!) of $\alpha $ by the correspondence $[\Gamma _ f]$ from $X$ to $Y$, see Lemma 45.3.6. Hence $f^!\alpha $ viewed as a correspondence from $\mathop{\mathrm{Spec}}(k)$ to $Y$ is equal to $a \circ [\Gamma _ f]$, see Lemma 45.3.1. Since $G$ is a functor, we conclude $\gamma $ is compatible with pullbacks, i.e., axiom (C)(a) holds.

Let $f : Y \to X$ be a morphism of smooth projective varieties and let $\beta \in \mathop{\mathrm{CH}}\nolimits ^ r(Y)$ be a cycle on $Y$. We have to show that

for all $c \in H^*(X)$. Let $a, a^ t, \eta _ X, \eta _ Y, [X], [Y]$ be as in Lemma 45.3.9. Let $b$ be $\beta $ viewed as a correspondence from $\mathop{\mathrm{Spec}}(k)$ to $Y$ of degree $r$. Then $f_*\beta $ viewed as a correspondence from $\mathop{\mathrm{Spec}}(k)$ to $X$ is equal to $a^ t \circ b$, see Lemmas 45.3.6 and 45.3.1. The displayed equality above holds if we can show that

is equal to

This follows immediately from Lemma 45.3.9. Thus we have axiom (C)(b).

To prove axiom (C)(c) we use the discussion in Remark 45.7.2. Hence it suffices to prove that $\gamma $ is compatible with exterior products. Let $X$, $Y$ be smooth projective varieties and let $\alpha $, $\beta $ be cycles on them. Denote $a$, $b$ the corresponding correspondences from $\mathop{\mathrm{Spec}}(k)$ to $X$, $Y$. Then $\alpha \times \beta $ corresponds to the correspondence $a \otimes b$ from $\mathop{\mathrm{Spec}}(k)$ to $X \otimes Y = X \times Y$. Hence the requirement follows from the fact that $G$ is compatible with the tensor structures on both sides.

Axiom (C)(d) follows because the cycle $[\mathop{\mathrm{Spec}}(k)]$ corresponds to the identity morphism on $h(\mathop{\mathrm{Spec}}(k))$. This finishes the proof of the lemma. $\square$

Lemma 45.7.10. Let $k$ be an algebraically closed field. Let $F$ be a field of characteristic $0$. Let $H^*$ be a classical Weil cohomology theory. Then we can construct a $\mathbf{Q}$-linear functor

of symmetric monoidal categories such that $H^*(X) = G(h(X))$.

**Proof.**
By Lemma 45.4.8 it suffices to construct a functor $G$ on the category of smooth projective schemes over $k$ with morphisms given by correspondences of degree $0$ such that the image of $G(c_2)$ on $G(\mathbf{P}^1)$ is an invertible graded $F$-vector space. Since every smooth projective scheme is canonically a disjoint union of smooth projective varieties, it suffices to construct $G$ on the category whose objects are smooth projective varieties and whose morphisms are correspondences of degree $0$. (Some details omitted.)

Given a smooth projective variety $X$ we set $G(X) = H^*(X)$.

Given a correspondence $c \in \text{Corr}^0(X, Y)$ between smooth projective varieties we consider the map $G(c) : G(X) = H^*(X) \to G(Y) = H^*(Y)$ given by the rule

It is clear that $G(c)$ is additive in $c$ and hence $\mathbf{Q}$-linear. Compatibility of $\gamma $ with pullbacks, pushforwards, and intersection products given by axioms (C)(a), (C)(b), and (C)(c) shows that we have $G(c' \circ c) = G(c') \circ G(c)$ if $c' \in \text{Corr}^0(Y, Z)$. Namely, for $a \in H^*(X)$ we have

with obvious notation. The first equality follows from the definitions. The second equality holds because $\text{pr}^{23, *}_2 \circ \text{pr}^{12}_{2, *} = \text{pr}^{123}_{23, *} \circ \text{pr}^{123, *}_{12}$ as follows immediately from the description of pushforward along projections given in Lemma 45.7.6. The third equality holds by Lemma 45.7.1 and the fact that $H^*$ is a functor. The fourth equalith holds by axiom (C)(a) and the fact that the gysin map agrees with flat pullback for flat morphisms (Chow Homology, Lemma 42.58.5). The fifth equality uses axiom (C)(c) as well as Lemma 45.7.1 to see that $\text{pr}^{23}_{3, *} \circ \text{pr}^{123}_{23, *} = \text{pr}^{13}_{3, *} \circ \text{pr}^{123}_{13, *}$. The sixth equality uses the projection formula from Lemma 45.7.1 as well as axiom (C)(b) to see that $ \text{pr}^{123}_{13, *} \gamma (\text{pr}^{123, *}_{23}c' \cdot \text{pr}^{123, *}_{12}c) = \gamma (\text{pr}^{123}_{13, *}( \text{pr}^{123, *}_{23}c' \cdot \text{pr}^{123, *}_{12}c))$. Finally, the last equality is the definition.

To finish the proof that $G$ is a functor, we have to show identities are preserved. In other words, if $1 = [\Delta ] \in \text{Corr}^0(X, X)$ is the identity in the category of correspondences (see Lemma 45.3.3 and its proof), then we have to show that $G([\Delta ]) = \text{id}$. This follows from the determination of $\gamma ([\Delta ])$ in Lemma 45.7.7 and Lemma 45.7.6. This finishes the construction of $G$ as a functor on smooth projective varieties and correspondences of degree $0$.

It follows from axioms (A)(c) and (A)(d) that $G(\mathop{\mathrm{Spec}}(k)) = H^*(\mathop{\mathrm{Spec}}(k))$ is canonically isomorphic to $F$ as an $F$-algebra. The Künneth axiom (B) shows our functor is compatible with tensor products. Thus our functor is a functor of symmetric monoidal categories.

We still have to check that the image of $G(c_2)$ on $G(\mathbf{P}^1)$ is an invertible graded $F$-vector space (in particular we don't know yet that $G$ extends to $M_ k$). By axiom (A)(d) the map $\int _{\mathbf{P}^1} : H^2(\mathbf{P}^1) \to F$ is an isomorphism. By axiom (A)(b) we see that $\dim _ F H^0(\mathbf{P}^1) = 1$. By Lemma 45.7.8 and axiom (A)(c) we obtain $2 - \dim _ F H^1(\mathbf{P}^1) = c_1(T_{\mathbf{P}^1}) = 2$. Hence $H^1(\mathbf{P}^1) = 0$. Thus

Recall that $1 = c_0 + c_2$ is a decomposition of the identity into a sum of orthogonal idempotents in $\text{Corr}^0(\mathbf{P}^1, \mathbf{P}^1)$, see Example 45.3.7. We have $c_0 = a \circ b$ where $a \in \text{Corr}^0(\mathop{\mathrm{Spec}}(k), \mathbf{P}^1)$ and $b \in \text{Corr}^0(\mathbf{P}^1, \mathop{\mathrm{Spec}}(k))$ and where $b \circ a = 1$ in $\text{Corr}^0(\mathop{\mathrm{Spec}}(k), \mathop{\mathrm{Spec}}(k))$, see proof of Lemma 45.4.4. Since $F = G(\mathop{\mathrm{Spec}}(k))$, it follows from functoriality that $G(c_0)$ is the projector onto the summand $H^0(\mathbf{P}^1) \subset G(\mathbf{P}^1)$. Hence $G(c_2)$ must necessarily be the projection onto $H^2(\mathbf{P}^1)$ and the proof is complete. $\square$

Proposition 45.7.11. Let $k$ be an algebraically closed field. Let $F$ be a field of characteristic $0$. A classical Weil cohomology theory is the same thing as a $\mathbf{Q}$-linear functor

of symmetric monoidal categories together with an isomorphism $F[2] \to G(\mathbf{1}(1))$ of graded $F$-vector spaces such that in addition

$G(h(X))$ lives in nonnegative degrees, and

$\dim _ F G^0(h(X)) = 1$

for any smooth projective variety $X$.

**Proof.**
Given $G$ and $F[2] \to G(\mathbf{1}(1))$ by setting $H^*(X) = G(h(X))$ we obtain data (D1), (D2), and (D3) satisfying all of (A), (B), and (C) except for possibly (A)(c) and (A)(d), see Lemma 45.7.9 and its proof. Observe that assumptions (1) and (2) imply axioms (A)(c) and (A)(d) in the presence of the known axioms (A)(a) and (A)(b).

Conversely, given $H^*$ we get a functor $G$ by the construction of Lemma 45.7.10. Let $X = \mathbf{P}^1, c_0, c_2$ be as in Example 45.3.7. We have constructed an isomorphism $1(-1) \to (X, c_2, 0)$ of motives in Lemma 45.4.4. In the proof of Lemma 45.7.10 we have seen that $G(1(-1)) = G(X, c_2, 0) = H^2(\mathbf{P}^1)[-2]$. Hence the isomorphism $\int _{\mathbf{P}^1} : H^2(\mathbf{P}^1) \to F$ of axiom (A)(d) gives an isomorphism $G(1(-1)) \to F[-2]$ which determines an isomorphism $F[2] \to G(\mathbf{1}(1))$. Finally, since $G(h(X)) = H^*(X)$ assumptions (1) and (2) follow from axiom (A). $\square$

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