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
\[ G : M_ k \longrightarrow \text{graded }F\text{-vector spaces} \]
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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