Definition 45.11.4. Let $k$ be a field. Let $F$ be a field of characteristic $0$. A *Weil cohomology theory* over $k$ with coefficients in $F$ is given by data (D0), (D1), (D2), and (D3) satisfying Poincaré duality, the Künneth formula, and compatibility with cycle classes, more precisely, satisfying axioms (A), (B), and (C) of Section 45.9 and in addition such that the equivalent conditions (1) and (2) of Lemma 45.11.2 hold for every smooth projective $X$ over $k$.

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