The following proposition is proved in EGA. Since EGA is somewhat intimidating(and it is written in French), I'm looking for a more readable proof.

EGA IV-3 (8.9.1) (ii) Let $A$ be a ring. Let $X$ be a scheme which is quasi-compact, quasi-separated and locally of finite presentation over $A$. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module of finite presentation. Then there exist a subring $A_0$ of $A$, which is of finite type over $\mathbb{Z}$ and a scheme $X_0$ of finite type over $A_0$ and a coherent $\mathcal{O}_{X_0}$-module $\mathcal{F}_0$ such that $X = X_0\times _{A_0} A$ and $\mathcal{F} = \mathcal{F}_0\otimes_{A_0} A$. Moreover $Supp(\mathcal{F})$ is constructible and closed, and there exists a closed subscheme $Z$ of $X$ such that $|Z| = Supp(\mathcal{F})$ and the canonical morphism $Z \rightarrow X$ is locally of finite presentation.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.