On invariants dual to the Bass numbers

Authors:
Edgar Enochs and Jinzhong Xu

Journal:
Proc. Amer. Math. Soc. **125** (1997), 951-960

MSC (1991):
Primary 13C11, 13E05

MathSciNet review:
1363457

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a commutative Noetherian ring, and let be an -module. In earlier papers by Bass (1963) and Roberts (1980) the Bass numbers were defined for all primes and all integers by use of the minimal injective resolution of . It is well known that . On the other hand, if is finitely generated, the *Betti* numbers are defined by the minimal free resolution of over the local ring . In an earlier paper of the second author (1995), using the flat covers of modules, the invariants were defined by the minimal flat resolution of over Gorenstein rings. The invariants were shown to be somehow dual to the Bass numbers. In this paper, we use homologies to compute these invariants and show that

for any cotorsion module . Comparing this with the computation of the Bass numbers, we see that is replaced by and the localization is replaced by (which was called the colocalization of at the prime ideal by Melkersson and Schenzel).

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Additional Information

**Edgar Enochs**

Affiliation:
Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506

**Jinzhong Xu**

Affiliation:
Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506

DOI:
https://doi.org/10.1090/S0002-9939-97-03662-9

Keywords:
Bass numbers,
minimal flat resolutions,
cotorsion modules

Received by editor(s):
February 22, 1995

Received by editor(s) in revised form:
August 16, 1995

Communicated by:
Wolmer V. Vasconcelos

Article copyright:
© Copyright 1997
American Mathematical Society