The Stacks project

Lemma 66.48.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_ X$-algebras. There exists a quasi-coherent sheaf of $\mathcal{O}_ X$-algebras $\mathcal{A}' \subset \mathcal{A}$ such that for any affine object $U$ of $X_{\acute{e}tale}$ the ring $\mathcal{A}'(U) \subset \mathcal{A}(U)$ is the integral closure of $\mathcal{O}_ X(U)$ in $\mathcal{A}(U)$.

Proof. Let $U$ be an object of $X_{\acute{e}tale}$. Then $U$ is a scheme. Denote $\mathcal{A}|_ U$ the restriction to the Zariski site. Then $\mathcal{A}|_ U$ is a quasi-coherent sheaf of $\mathcal{O}_ U$-algebras hence we can apply Morphisms, Lemma 29.53.1 to find a quasi-coherent subalgebra $\mathcal{A}'_ U \subset \mathcal{A}|_ U$ such that the value of $\mathcal{A}'_ U$ on any affine open $W \subset U$ is as given in the statement of the lemma. If $f : U' \to U$ is a morphism in $X_{\acute{e}tale}$, then $\mathcal{A}|_{U'} = f^*(\mathcal{A}|_ U)$ where $f^*$ means pullback by the morphism $f$ in the Zariski topology; this holds because $\mathcal{A}$ is quasi-coherent (see introduction to Properties of Spaces, Section 65.29 and the references to the discussion in the chapter on descent on schemes). Since $f$ is ├ętale we find that More on Morphisms, Lemma 37.19.1 says that we get a canonical isomorphism $f^*(\mathcal{A}'_ U) = \mathcal{A}'_{U'}$. This immediately tells us that we obtain a sub presheaf $\mathcal{A}' \subset \mathcal{A}$ of $\mathcal{O}_ X$-algebras over $X_{\acute{e}tale}$ which is a sheaf for the Zariski topology and has the right values on affine objects. But the fact that each $\mathcal{A}'_ U$ is quasi-coherent on the scheme $U$ and that for $f : U' \to U$ ├ętale we have $\mathcal{A}'_{U'} = f^*(\mathcal{A}'_ U)$ implies that $\mathcal{A}'$ is quasi-coherent on $X_{\acute{e}tale}$ as well (as this is a local property and we have the references above describing quasi-coherent modules on $U_{\acute{e}tale}$ in exactly this manner). $\square$


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