Proposition 45.9.12 (0FHM)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 45: Weil Cohomology Theories
Section 45.9: Weil cohomology theories, I
Proposition 45.9.12 (cite)

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Proposition 45.9.12. Let $k$ be a field. Let $F$ be a field of characteristic $0$. There is a $1$-to-$1$ correspondence between the following

data (D0), (D1), (D2), and (D3) satisfying (A), (B), and(C), and
$\mathbf{Q}$-linear symmetric monoidal functors

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

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

Proof. Given $G$ as in (2) by setting $H^*(X) = G(h(X))$ we obtain data (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C), see Lemma 45.9.10 and its proof.

Conversely, given data (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C) we get a functor $G$ as in (2) by the construction of the proof of Lemma 45.9.11.

We omit the detailed proof that these constructions are inverse to each other. $\square$

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