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Proceedings of the American Mathematical Society

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

 
 

 

The condition $\textrm {Ext}^1(M, R) = 0$ 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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Keywords: Artin local algebra, reflexive, Cohen-Macaulay, Gorenstein
Article copyright: © Copyright 1974 American Mathematical Society