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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.

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up vote 3 down vote accepted

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.

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Although my question as stated is more general, all schemes in Milne's book are locally Noetherian so I am more than happy with this answer. –  Joel Dodge Apr 11 '10 at 5:13
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This works for any qcqs scheme, replacing "coherent" with "qcoh of finite type". It is in EGA I, around 9.1.6 or so, in qc septd case and in noetherian case; generalization to qcqs given in erratum in later EGA (probably IV_1). As for assuming "all schemes are locally noetherian", this is not always such a good idea; rules out natural tensor product operations with non-finite-type maps, such as henselization or completion tensored against itself over base ring (and much more); the henselian example comes up in proof of smooth base change thm, and something like it arises in Milne's book too. –  BCnrd Apr 11 '10 at 12:23
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