Proposition 45.11.6. Let $k$ be a field. Let $F$ be a field of characteristic $0$. A Weil cohomology theory is the same thing as a $\mathbf{Q}$-linear symmetric monoidal functor

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

such that

1. $G(\mathbf{1}(1))$ is nonzero only in degree $-2$, and

2. for every smooth projective scheme $X$ over $k$ with $k' = \Gamma (X, \mathcal{O}_ X)$ the homomorphism $G(h(\mathop{\mathrm{Spec}}(k'))) \to G(h(X))$ of graded $F$-vector spaces is an isomorphism in degree $0$.

Proof. Immediate consequence of Proposition 45.9.12 and Definition 45.11.4. Of course we could replace (2) by the condition that $G(h(X)) \to \bigoplus G(h(x_ i))$ is injective in degree $0$ for some choice of closed points $x_1, \ldots , x_ r \in X$ whose residue fields are separable over $k$. $\square$

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