The condition for modules over local Artin algebras
with
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 | References | Similar Articles | Additional Information
Abstract: Let be a finitely generated module over a (not necessarily commutative) local Artin algebra
with
. It is known that when
is Gorenstein (i.e. of finite injective dimension)
. For
not Gorenstein we describe all
with
and show that
for some
if and only if
is free. It follows that for
not Gorenstein all reflexives are free. We also calculate the lengths of all the
. As an application we show that if
is a commutative Cohen-Macaulay local ring of dimension
which is not Gorenstein, if
is Artin and
is a system of parameters with
contained in the ideal
and if
is a finitely generated
-module with
for
, then
is free.
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- [2] Maurice Auslander, Comments on the functor 𝐸𝑥𝑡, Topology 8 (1969), 151–166. MR 237606, https://doi.org/10.1016/0040-9383(69)90006-8
- [3] Hyman Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488. MR 157984, https://doi.org/10.1090/S0002-9947-1960-0157984-8
- [4] Anneaux de Gorenstein, et torsion en algèbre commutative, Séminaire d’Algèbre Commutative dirigé par Pierre Samuel, 1966/67. Texte rédigé, d’après des exposés de Maurice Auslander, Marguerite Mangeney, Christian Peskine et Lucien Szpiro. École Normale Supérieure de Jeunes Filles, Secrétariat mathématique, Paris, 1967 (French). MR 0225844
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Additional Information
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