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

such that

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

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 $0$.

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