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On invariants dual to the Bass numbers

Authors:	Edgar Enochs Jinzhong Xu

Institution:	Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506 Jinzhong Xu ; Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506

Abstract:	Let be a commutative Noetherian ring, and let be an -module. In earlier papers by Bass (1963) and Roberts (1980) the Bass numbers $\mu _i(p,M)$ were defined for all primes and all integers $i\ge 0$ by use of the minimal injective resolution of . 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 is finitely generated, the Betti numbers $\beta _i(p,M)$ 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 $\pi _i(p,M)$ were defined by the minimal flat resolution of 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 . Comparing this with the computation of the Bass numbers, we see that $\operatorname {Ext}$ is replaced by $\operatorname {Tor}$ and the localization is replaced by $\operatorname {Hom}_R(R_p,M)$ (which was called the colocalization of at the prime ideal by Melkersson and Schenzel).

Keywords:	Bass numbers minimal flat resolutions cotorsion modules

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