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Proceedings of the American Mathematical Society

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



On the vanishing of $ {\rm Ext}$

Author: Mark Ramras
Journal: Proc. Amer. Math. Soc. 27 (1971), 457-462
MSC: Primary 13.40
MathSciNet review: 0284427
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Abstract: In this paper we exhibit certain modules $ A$ over a commutative noetherian local ring $ (R,\mathfrak{M})$ which test projective dimension of finitely generated modules in the following sense: if $ {\operatorname{Ext} ^j}(M,A) = 0$ for all $ j \geqq i$, then pd $ M < i$.

We also show that the module $ \mathfrak{M}$ tests in a stronger way: if $ {\operatorname{Ext} ^i}(M,\mathfrak{M}) = 0$, then pd $ M < i$.

In conclusion we show that if $ (R,\mathfrak{M})$ is artin, then $ R$ is self-injective if and only if $ {\operatorname{Ext} ^1}(R/{\mathfrak{M}^n},R) = 0$, where the index of nilpotence of $ \mathfrak{M}$ is $ n + 1$.

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Keywords: Projective dimension, Ext, commutative noetherian local ring, length of a module, artin local ring, socle
Article copyright: © Copyright 1971 American Mathematical Society

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