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Is there any text that gives some applications of sheaves theory in commutative ring theory? In the other word, is any results in commutative ring theory that be verified by sheaves method?

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    $\begingroup$ Um, someone can correct me if I'm wrong, but I think the key words here are "algebraic geometry". $\endgroup$
    – Paul Siegel
    Commented Nov 20, 2015 at 11:23

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The ring $\mathbb{C}\{z_1,\dots,z_n\}^\dagger$ of functions that are holomorphic in some neighborhood of the closed unit disk in $\mathbb{C}^n$ is noetherian.

This is a theorem of J. Frisch (Points de platitude d’un morphisme d’espaces analytiques complexes, Invent. Math. 4 (1967), p. 118–138). Except for $n=1$, I do not know of any proof that does not use sheaf theory quite heavily: coherence of the structure sheaf of $\mathbb{C}^n$, vanishing of coherent cohomology on closed disks, etc.

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The whole book of Pierce, "modules over commutative regular rings" is devoted to sheaf theory of rings.

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