The Stacks project

Lemma 46.5.2. Let $S$ be a scheme. Let $\mathcal{F}$ be an adequate $\mathcal{O}$-module on $(\mathit{Sch}/S)_\tau $. For any affine scheme $\mathop{\mathrm{Spec}}(A)$ over $S$ the functor $F_{\mathcal{F}, A}$ is adequate.

Proof. Let $\{ \mathop{\mathrm{Spec}}(A_ i) \to S\} _{i \in I}$ be a $\tau $-covering such that $F_{\mathcal{F}, A_ i}$ is adequate for all $i \in I$. We can find a standard affine $\tau $-covering $\{ \mathop{\mathrm{Spec}}(A'_ j) \to \mathop{\mathrm{Spec}}(A)\} _{j = 1, \ldots , m}$ such that $\mathop{\mathrm{Spec}}(A'_ j) \to \mathop{\mathrm{Spec}}(A) \to S$ factors through $\mathop{\mathrm{Spec}}(A_{i(j)})$ for some $i(j) \in I$. Then we see that $F_{\mathcal{F}, A'_ j}$ is the restriction of $F_{\mathcal{F}, A_{i(j)}}$ to the category of $A'_ j$-algebras. Hence $F_{\mathcal{F}, A'_ j}$ is adequate by Lemma 46.3.17. By Lemma 46.3.19 the sequence $F_{\mathcal{F}, A'_ j}$ corresponds to an adequate “product” functor $F'$ over $A' = A'_1 \times \ldots \times A'_ m$. As $\mathcal{F}$ is a sheaf (for the Zariski topology) this product functor $F'$ is equal to $F_{\mathcal{F}, A'}$, i.e., is the restriction of $F$ to $A'$-algebras. Finally, $\{ \mathop{\mathrm{Spec}}(A') \to \mathop{\mathrm{Spec}}(A)\} $ is a $\tau $-covering. It follows from Lemma 46.3.20 that $F_{\mathcal{F}, A}$ is adequate. $\square$

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