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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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: Let $R$ be a commutative Noetherian ring, and let $M$ be an $R$-module. In earlier papers by Bass (1963) and Roberts (1980) the Bass numbers $\mu _i(p,M)$ were defined for all primes $p$ and all integers $i\ge 0$ by use of the minimal injective resolution of $M$. It is well known that $\mu _i(p,M)=\dim _{k(p)}\operatorname {Ext} _{R_p}^i(k(p),M_p)$. On the other hand, if $M$ is finitely generated, the Betti numbers $\beta _i(p,M)$ are defined by the minimal free resolution of $M_p$ over the local ring $R_p$. In an earlier paper of the second author (1995), using the flat covers of modules, the invariants $\pi _i(p,M)$ were defined by the minimal flat resolution of $M$ over Gorenstein rings. The invariants $\pi _i(p,M)$ were shown to be somehow dual to the Bass numbers. In this paper, we use homologies to compute these invariants and show that

\begin{displaymath}\pi _i(p,M)= \dim _{k(p)}\operatorname {Tor}_i^{R_p}(k(p), \operatorname {Hom}_R(R_p,M))\end{displaymath}

for any cotorsion module $M$. Comparing this with the computation of the Bass numbers, we see that $\operatorname {Ext}$ is replaced by $\operatorname {Tor}$ and the localization $M_p$ is replaced by $\operatorname {Hom}_R(R_p,M)$ (which was called the colocalization of $M$ at the prime ideal $p$ 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: http://dx.doi.org/10.1090/S0002-9939-97-03662-9
PII: S 0002-9939(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