The Stacks project

Lemma 51.19.3. Let $A$ be a ring. Let $V \to \mathop{\mathrm{Spec}}(A)$ be quasi-compact, quasi-separated, and étale. For each derivation $\theta : A \to A$ there exists a canonical additive operator $D$ on $H^ i(V, \mathcal{O}_ V)$ satisfying the Leibniz rule with respect to $\theta $.

Proof. If $V$ is separated, then we can argue using an affine open covering $V = \bigcup _{j = 1, \ldots m} V_ j$. Namely, because $V$ is separated we may write $V_{j_0 \ldots j_ p} = \mathop{\mathrm{Spec}}(B_{j_0 \ldots j_ p})$. See Schemes, Lemma 26.21.7. Then we find that the $A$-module $H^ i(V, \mathcal{O}_ V)$ is the $i$th cohomology group of the Čech complex

\[ \prod B_{j_0} \to \prod B_{j_0j_1} \to \prod B_{j_0j_1j_2} \to \ldots \]

See Cohomology of Schemes, Lemma 30.2.6. Each $B = B_{j_0 \ldots j_ p}$ is an étale $A$-algebra. Hence $\Omega _ B = \Omega _ A \otimes _ A B$ and we conclude $\theta $ extends uniquely to a derivation $\theta _ B : B \to B$. These maps define an endomorphism of the Čech complex and define the desired operators on the cohomology groups.

In the general case we use a hypercovering of $V$ by affine opens, exactly as in the first part of the proof of Cohomology of Schemes, Lemma 30.7.3. We omit the details. $\square$

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