## On invariants dual to the Bass numbers

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- by Edgar Enochs and Jinzhong Xu
- Proc. Amer. Math. Soc.
**125**(1997), 951-960 - DOI: https://doi.org/10.1090/S0002-9939-97-03662-9
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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 \[ \pi _i(p,M)= \dim _{k(p)}\operatorname {Tor}_i^{R_p}(k(p), \operatorname {Hom}_R(R_p,M))\] 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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## Bibliographic 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
- Received by editor(s): February 22, 1995
- Received by editor(s) in revised form: August 16, 1995
- Communicated by: Wolmer V. Vasconcelos
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**125**(1997), 951-960 - MSC (1991): Primary 13C11, 13E05
- DOI: https://doi.org/10.1090/S0002-9939-97-03662-9
- MathSciNet review: 1363457