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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).

References [Enhancements On Off] (What's this?)

  • 1. M. Auslander and I. Reiten, Applications of contravariantly finite subcategories, Adv. Math. 86 (1991), 111-152. MR 92e:16009
  • 2. M. Auslander and R. Buchweitz, The homological theory of maximal Cohen-Macaulay approximations, Soc. Math. de France, Memoire 38 (1989), 5-37. MR 91h:13010
  • 3. H. Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8-28. MR 27:3669
  • 4. E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. of Math. 39 (1981), 33-38. MR 83a:16031
  • 5. --, Flat covers and flat cotorsion modules, Proc. Amer. Math. Soc. 92 (1984), 179-184. MR 85j:13016
  • 6. --, Minimal pure injective resolutions of flat modules, J. of Algebra 105 (2) (1987), 351-364. MR 88f:13001
  • 7. E. Matlis, Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511-528. MR 20:5800
  • 8. Leif Melkersson and Peter Schenzel, The co-localization of an Artinian module, Proc. Edinburgh Math. Soc. (2) 38 (1995), 121-131. MR 96a:13020
  • 9. P. Roberts, Homological invariants of modules over commutative rings, Les Presses de l'Univ. de Montreal, 1980. MR 82j:13020
  • 10. Jinzhong Xu, Minimal injective and flat resolutions of modules over Gorenstein rings, J. of Algebra 175 (1995), 451-477. MR 96h:13025
  • 11. --, The existence of flat covers over Noetherian rings of finite Krull dimension, Proc. Amer. Math. Soc. 123 (1) (1995), 27-32. MR 95c:16004

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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

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

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