Bhatwadekar, S. M. Analytic isomorphism and category of finitely generated modules. (English) Zbl 0656.13018 Commun. Algebra 16, No. 9, 1949-1958 (1988). [Note: Some diagrams below cannot be displayed correctly in the web version. Please use the PDF version for correct display.]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 Cited in 1 ReviewCited in 6 Documents MSC: 13E15 Commutative rings and modules of finite generation or presentation; number of generators 13E05 Commutative Noetherian rings and modules 18B99 Special categories Keywords:noetherian rings; analytic isomorphism; categories of finitely generated modules; Bass-Quillen conjecture PDFBibTeX XMLCite \textit{S. M. Bhatwadekar}, Commun. Algebra 16, No. 9, 1949--1958 (1988; Zbl 0656.13018) Full Text: DOI References: [1] DOI: 10.1090/S0002-9947-1983-0709584-1 · doi:10.1090/S0002-9947-1983-0709584-1 [2] DOI: 10.1007/BF01389017 · Zbl 0477.13006 · doi:10.1007/BF01389017 [3] DOI: 10.1007/BF01223888 · Zbl 0351.13006 · doi:10.1007/BF01223888 [4] Mohan Kumar N., J.Indian Math.Soc 43 pp 13– (1979) [5] DOI: 10.1016/0022-4049(82)90050-0 · Zbl 0484.13008 · doi:10.1016/0022-4049(82)90050-0 [6] DOI: 10.1080/00927878108822596 · Zbl 0453.20042 · doi:10.1080/00927878108822596 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.