The Stacks project

Lemma 28.51.1. Let $X$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_ X$-algebras. The subsheaf $\mathcal{A}' \subset \mathcal{A}$ defined by the rule

\[ U \longmapsto \{ f \in \mathcal{A}(U) \mid f_ x \in \mathcal{A}_ x \text{ integral over } \mathcal{O}_{X, x} \text{ for all }x \in U\} \]

is a quasi-coherent $\mathcal{O}_ X$-algebra, the stalk $\mathcal{A}'_ x$ is the integral closure of $\mathcal{O}_{X, x}$ in $\mathcal{A}_ x$, and for any affine open $U \subset X$ the ring $\mathcal{A}'(U) \subset \mathcal{A}(U)$ is the integral closure of $\mathcal{O}_ X(U)$ in $\mathcal{A}(U)$.

Proof. This is a subsheaf by the local nature of the conditions. It is an $\mathcal{O}_ X$-algebra by Algebra, Lemma 10.35.7. Let $U \subset X$ be an affine open. Say $U = \mathop{\mathrm{Spec}}(R)$ and say $\mathcal{A}$ is the quasi-coherent sheaf associated to the $R$-algebra $A$. Then according to Algebra, Lemma 10.35.12 the value of $\mathcal{A}'$ over $U$ is given by the integral closure $A'$ of $R$ in $A$. This proves the last assertion of the lemma. To prove that $\mathcal{A}'$ is quasi-coherent, it suffices to show that $\mathcal{A}'(D(f)) = A'_ f$. This follows from the fact that integral closure and localization commute, see Algebra, Lemma 10.35.11. The same fact shows that the stalks are as advertised. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 035F. Beware of the difference between the letter 'O' and the digit '0'.