Quasi-coherent sheaves of O_X-algebras

2010-04-11T03:35:32Z

Let $X$ be a quasi-compact scheme and let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_X$-algebras on $X$. $X$ being quasi-compact, we can write $X = U_1 \cup \dots \cup U_n$ with each $U_i$ affine, say $U_i =\textrm{Spec}R_i$, and such that $\mathcal{A}\mid_{U_i} \simeq \widetilde{A_i}$ for some finitely generated $R_i$ algebras $A_i$. Suppose we know that each $A_i$ is the union of its subalgebras which are module finite over $R_i$. Can we say that $\mathcal{A}$ is globally the union of it's subsheaves which are coherent sheaves of $\mathcal{O}_X$-algebras?

It seems that the obvious thing to do would be to glue together coherent algebras over each of the $R_i$'s, but it's not clear to me how this can be done. This question arose from Milne's proof of 'Zariski's Main Theorem' from the beginning of his etale cohomology book.

Answer by Emerton for Quasi-coherent sheaves of O_X-algebras

2010-04-11T04:55:25Z

Under some slightly stronger hypothesis (Noetherian is certainly enough) we may write $\mathcal A$ as the union of its coherent subsheaves. If $\mathcal E$ is a coherent subsheaf, then the subalgebra of $\mathcal A$ that it generates will also be coherent, because this can be tested locally, where it then follows from your assumptions. Thus in this case, $\mathcal A$ is the union of coherent $\mathcal O_X$-algebras.

I'm not sure how good a notion coherent is outside of the Noetherian context. If no-one gives an answer in the non-Noetherian context, then you might want to look at the stacks project, which discusses this kind of "coherent approximation to quasi-coherent sheaves" in some generality, if I remember correctly.

Quasi-coherent sheaves of O_X-algebras - MathOverflow

Quasi-coherent sheaves of O_X-algebras

Answer by Emerton for Quasi-coherent sheaves of O_X-algebras