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Proceedings of the American Mathematical Society

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

 
 

 

The condition $ {\rm Ext}^1(M, R) = 0$ for modules over local Artin algebras $ (R, \mathfrak{M})$ with $ \mathfrak{M}^2 = 0$


Author: Margaret S. Menzin
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
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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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DOI: https://doi.org/10.1090/S0002-9939-1974-0330227-3
Keywords: Artin local algebra, reflexive, Cohen-Macaulay, Gorenstein
Article copyright: © Copyright 1974 American Mathematical Society

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