On invariants dual to the Bass numbers

Enochs, Edgar; Xu, Jinzhong

doi:10.1090/S0002-9939-97-03662-9

On invariants dual to the Bass numbers
HTML articles powered by AMS MathViewer

by Edgar Enochs and Jinzhong Xu PDF

Proc. Amer. Math. Soc. 125 (1997), 951-960 Request permission

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).

References

Maurice Auslander and Idun Reiten, Applications of contravariantly finite subcategories, Adv. Math. 86 (1991), no. 1, 111–152. MR 1097029, DOI 10.1016/0001-8708(91)90037-8
Maurice Auslander and Ragnar-Olaf Buchweitz, The homological theory of maximal Cohen-Macaulay approximations, Mém. Soc. Math. France (N.S.) 38 (1989), 5–37 (English, with French summary). Colloque en l’honneur de Pierre Samuel (Orsay, 1987). MR 1044344
Hyman Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8–28. MR 153708, DOI 10.1007/BF01112819
Edgar E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981), no. 3, 189–209. MR 636889, DOI 10.1007/BF02760849
Edgar Enochs, Flat covers and flat cotorsion modules, Proc. Amer. Math. Soc. 92 (1984), no. 2, 179–184. MR 754698, DOI 10.1090/S0002-9939-1984-0754698-X
Edgar E. Enochs, Minimal pure injective resolutions of flat modules, J. Algebra 105 (1987), no. 2, 351–364. MR 873670, DOI 10.1016/0021-8693(87)90200-6
Eben Matlis, Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511–528. MR 99360, DOI 10.2140/pjm.1958.8.511
Leif Melkersson and Peter Schenzel, The co-localization of an Artinian module, Proc. Edinburgh Math. Soc. (2) 38 (1995), no. 1, 121–131. MR 1317331, DOI 10.1017/S0013091500006258
Paul Roberts, Homological invariants of modules over commutative rings, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 72, Presses de l’Université de Montréal, Montreal, Que., 1980. MR 569936
Jin Zhong Xu, Minimal injective and flat resolutions of modules over Gorenstein rings, J. Algebra 175 (1995), no. 2, 451–477. MR 1339651, DOI 10.1006/jabr.1995.1196
Jin Zhong Xu, The existence of flat covers over Noetherian rings of finite Krull dimension, Proc. Amer. Math. Soc. 123 (1995), no. 1, 27–32. MR 1242111, DOI 10.1090/S0002-9939-1995-1242111-5