The Stacks project

Proof. This proof is the same as the proof of Lemma 45.7.9; we urge the reader to read the proof of that lemma instead.

We obtain a contravariant functor from the category of smooth projective schemes over $k$ 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 $-2$. Our datum (D0) is the vector space $F(1) = G^{-2}(\mathbf{1}(1))$. Since $G$ is a symmetric monoidal functor we see that $F(n) = G^{-2n}(\mathbf{1}(n))$ for all $n \in \mathbf{Z}$. It follows that

a formula we will frequently use below.

Let $X$ be a smooth projective scheme over $k$. By Lemma 45.3.1 we have

Applying the functor $G$ this maps into $\mathop{\mathrm{Hom}}\nolimits (G(\mathbf{1}), G(h(X)(r)))$. By taking the image of $1$ in $G^0(\mathbf{1}) = F$ into $G^0(h(X)(r)) = H^{2r}(X)(r)$ we obtain

This is the datum (D2).

Let $X$ be a nonempty smooth projective scheme over $k$ which is equidimensional 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 $G$ and taking degree $0$ parts we obtain

This map $\int _ X : H^{2d}(X)(d) \to F$ is the datum (D3).

Let $X$ be a smooth projective scheme over $k$ which is nonempty and equidimensional of dimension $d$. By Lemma 45.4.9 we know that $h(X)(d)$ is a left dual to $h(X)$. Hence $G(h(X)(d)) = H^*(X) \otimes _ F F(d)[2d]$ is a left dual to $H^*(X)$ in the category of graded $F$-vector spaces. Here $[n]$ is the shift functor on graded 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 $H^{2d - i}(X)(d) \otimes _ F H^ i(X) \to F$. 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).

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 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 nonempty equidimensional smooth projective schemes over $k$, 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 nonempty equidimensional smooth projective schemes over $k$ 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 nonempty smooth projective schemes over $k$ 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. $\square$