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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The condition $\textrm {Ext}^1(M, R) = 0$ for modules over local Artin algebras $(R, \mathfrak {M})$ with $\mathfrak {M}^2 = 0$
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by Margaret S. Menzin PDF
Proc. Amer. Math. Soc. 43 (1974), 47-52 Request permission

Abstract:

Let $M$ be a finitely generated module over a (not necessarily commutative) local Artin algebra $(R,\mathfrak {M})$ with ${\mathfrak {M}^2} = 0$. It is known that when $R$ is Gorenstein (i.e. of finite injective dimension) $M = \sum R \oplus \sum R/\mathfrak {M}$. For $R$ not Gorenstein we describe all $M$ with ${\operatorname {Ext} ^1}(M,R) = 0$ and show that ${\operatorname {Ext} ^i}(M,R) = 0$ for some $i > 1$ if and only if $M$ is free. It follows that for $R$ not Gorenstein all reflexives are free. We also calculate the lengths of all the ${\operatorname {Ext} ^i}(M,R)$. As an application we show that if $(R,\mathfrak {M})$ is a commutative Cohen-Macaulay local ring of dimension $d$ which is not Gorenstein, if $R/{\mathfrak {M}^2}$ is Artin and $({x_1}, \cdots ,{x_d})$ is a system of parameters with ${\mathfrak {M}^2}$ contained in the ideal $({x_1}, \cdots ,{x_d})$ and if $M$ is a finitely generated $R$-module with ${\operatorname {Ext} ^i}(M,R) = 0$ for $1 \leqq i \leqq 2d + 2$, then $M$ is free.
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Additional Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 43 (1974), 47-52
  • MSC: Primary 16A62
  • DOI: https://doi.org/10.1090/S0002-9939-1974-0330227-3
  • MathSciNet review: 0330227