The Stacks project

Lemma 46.5.6. Let $\mathcal{F}$ be an adequate $\mathcal{O}$-module on $(\mathit{Sch}/S)_\tau $. For any surjective flat morphism $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ of affines over $S$ the extended Čech complex

\[ 0 \to \mathcal{F}(\mathop{\mathrm{Spec}}(A)) \to \mathcal{F}(\mathop{\mathrm{Spec}}(B)) \to \mathcal{F}(\mathop{\mathrm{Spec}}(B \otimes _ A B)) \to \ldots \]

is exact. In particular $\mathcal{F}$ satisfies the sheaf condition for fpqc coverings, and is a sheaf of $\mathcal{O}$-modules on $(\mathit{Sch}/S)_{fppf}$.

Proof. With $A \to B$ as in the lemma let $F = F_{\mathcal{F}, A}$. This functor is adequate by Lemma 46.5.2. By Lemma 46.3.5 since $A \to B$, $A \to B \otimes _ A B$, etc are flat we see that $F(B) = F(A) \otimes _ A B$, $F(B \otimes _ A B) = F(A) \otimes _ A B \otimes _ A B$, etc. Exactness follows from Descent, Lemma 35.3.6.

Thus $\mathcal{F}$ satisfies the sheaf condition for $\tau $-coverings (in particular Zariski coverings) and any faithfully flat covering of an affine by an affine. Arguing as in the proofs of Descent, Lemma 35.5.1 and Descent, Proposition 35.5.2 we conclude that $\mathcal{F}$ satisfies the sheaf condition for all fpqc coverings (made out of objects of $(\mathit{Sch}/S)_\tau $). Details omitted. $\square$

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