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Analytic isomorphism and category of finitely generated modules. (English) Zbl 0656.13018

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Let S, T be commutative noetherian rings. A homomorphism \(f:\quad S\to T\) is called an analytic isomorphism along \(s\in S\) if the commutative diagram \(\begin{tikzcd} S \ar[r,"f"]\dar & T\dar \\ S_s \ar[r,"f_s" '] & T_{f(s)} \end{tikzcd}\) is cartesian, or, equivalently, if \(f\) induces and isomorphism \(S/s^ n S \cong T/f(s^ n)T\) for every positive integer n. Under this assumption the author proves that the induced diagram of the categories of finitely generated modules \(\begin{tikzcd} \text{mod-}S \rar\dar & \text{mod-}T \dar\\\text{mod-}S_s \rar & \text{mod-}T_{f(s)} \end{tikzcd} \) is also a cartesian square. This holds also for the categories of finite generated modules. The latter is already known in special cases and was used in context with the treatment of the Bass-Quillen conjecture on the extension of projective modules.
Reviewer: W.Grölz

MSC:

13E15 Commutative rings and modules of finite generation or presentation; number of generators
13E05 Commutative Noetherian rings and modules
18B99 Special categories
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