Bass and Betti numbers of a module and its deficiency modules. (English) Zbl 07866053
Let \(R\) denote a Gorenstein ring with \(d = \dim R \). Let \(M\) denote a Cohen-Macaulay \(R\)-module with \(t = \dim M\). Then H.-B. Foxby [Math. Scand. 29, 175–186 (1972; Zbl 0235.13006)] proved a duality between the Bass and Betti numbers of \(M\) and \(\operatorname{Ext}_R^{d-t}(M,R)\). Moreover, \(K^j(M) = \operatorname{Ext}_R^{d-j}(M,R)\) is called the \(j\)-th module of deficiency of \(M\). In particular \(K(M) = K^t(M)\) is called the canonical module of \(M\) (see [P. Schenzel, Prog. Math. 166, 241–292 (1998; Zbl 0949.13012)]). The authors study relations between Bass and Betti numbers of a given module and its deficiency modules. They establish relations in the case of generalized Cohen-Macaulay and canonically Cohen-Macaulay modules over local rings which are factors of Gorenstein rings. Besides of these generalizations they prove estimates for the Betti and Bass numbers of modules in terms of their deficiency modules. As a technical tool they use a spectral sequence invented by Foxby. Among further applications, a particular case of the Auslander-Reiten conjecture is shown.
Reviewer: Peter Schenzel (Halle an der Saale)
MSC:
| 13C14 | Cohen-Macaulay modules |
| 13D45 | Local cohomology and commutative rings |
| 13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |
| 14B15 | Local cohomology and algebraic geometry |
Keywords:
generalized Cohen-Macaulay module; homological dimensions; deficiency modules; Auslander-Reiten conjectureReferences:
| [1] | T. Araya, “The Auslander-Reiten conjecture for Gorenstein rings”, Proc. Amer. Math. Soc. 137:6 (2009), 1941-1944. Digital Object Identifier: 10.1090/S0002-9939-08-09757-8 Google Scholar: Lookup Link zbMATH: 1163.13015 · Zbl 1163.13015 · doi:10.1090/S0002-9939-08-09757-8 |
| [2] | T. Araya and Y. Yoshino, “Remarks on a depth formula, a grade inequality and a conjecture of Auslander”, Comm. Algebra 26:11 (1998), 3793-3806. Digital Object Identifier: 10.1080/00927879808826375 Google Scholar: Lookup Link zbMATH: 0906.13002 · Zbl 0906.13002 · doi:10.1080/00927879808826375 |
| [3] | M. Auslander and I. Reiten, “On a generalized version of the Nakayama conjecture”, Proc. Amer. Math. Soc. 52 (1975), 69-74. Digital Object Identifier: 10.2307/2040102 Google Scholar: Lookup Link · Zbl 0337.16004 · doi:10.2307/2040102 |
| [4] | L. L. Avramov and R.-O. Buchweitz, “Support varieties and cohomology over complete intersections”, Invent. Math. 142:2 (2000), 285-318. Digital Object Identifier: 10.1007/s002220000090 Google Scholar: Lookup Link · Zbl 0999.13008 · doi:10.1007/s002220000090 |
| [5] | M. P. Brodmann and L. T. Nhan, “On canonical Cohen-Macaulay modules”, J. Algebra 371 (2012), 480-491. Digital Object Identifier: 10.1016/j.jalgebra.2012.09.002 Google Scholar: Lookup Link · Zbl 1273.13026 · doi:10.1016/j.jalgebra.2012.09.002 |
| [6] | M. P. Brodmann and R. Y. Sharp, Local cohomology: An algebraic introduction with geometric applications, 2nd ed., Cambridge Studies in Advanced Mathematics 136, Cambridge Univ. Press, 2013. Digital Object Identifier: 10.1017/CBO9781139044059 Google Scholar: Lookup Link · doi:10.1017/CBO9781139044059 |
| [7] | W. Bruns and J. Herzog, Cohen-Macaulay rings, 2nd ed., Cambridge Studies in Advanced Mathematics 39, Cambridge Univ. Press, 1998. Digital Object Identifier: 10.1017/CBO9780511608681 Google Scholar: Lookup Link · Zbl 0909.13005 · doi:10.1017/CBO9780511608681 |
| [8] | H. Dao, M. Eghbali, and J. Lyle, “Hom and Ext, revisited”, J. Algebra 571 (2021), 75-93. Digital Object Identifier: 10.1016/j.jalgebra.2018.12.006 Google Scholar: Lookup Link · Zbl 1471.13026 · doi:10.1016/j.jalgebra.2018.12.006 |
| [9] | H.-B. Foxby, “On the \(\mu^i\) in a minimal injective resolution”, Math. Scand. 29 (1971), 175-186. Digital Object Identifier: 10.7146/math.scand.a-11043 Google Scholar: Lookup Link · Zbl 0235.13006 · doi:10.7146/math.scand.a-11043 |
| [10] | T. H. Freitas and V. H. Jorge Pérez, “When does the canonical module of a module have finite injective dimension?”, Arch. Math. (Basel) 117:5 (2021), 485-494. Digital Object Identifier: 10.1007/s00013-021-01659-0 Google Scholar: Lookup Link · Zbl 1471.13035 · doi:10.1007/s00013-021-01659-0 |
| [11] | M. Hashimoto, Auslander-Buchweitz approximations of equivariant modules, London Mathematical Society Lecture Note Series 282, Cambridge Univ. Press, 2000. Digital Object Identifier: 10.1017/CBO9780511565762 Google Scholar: Lookup Link · Zbl 0993.13007 · doi:10.1017/CBO9780511565762 |
| [12] | C. Huneke and G. J. Leuschke, “On a conjecture of Auslander and Reiten”, J. Algebra 275:2 (2004), 781-790. Digital Object Identifier: 10.1016/j.jalgebra.2003.07.018 Google Scholar: Lookup Link · Zbl 1096.13011 · doi:10.1016/j.jalgebra.2003.07.018 |
| [13] | J. Lyle and J. Montaño, “Extremal growth of Betti numbers and trivial vanishing of (co)homology”, Trans. Amer. Math. Soc. 373:11 (2020), 7937-7958. Digital Object Identifier: 10.1090/tran/8189 Google Scholar: Lookup Link · Zbl 1451.13054 · doi:10.1090/tran/8189 |
| [14] | H. Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics 8, Cambridge Univ. Press, 1989. Digital Object Identifier: 10.1017/CBO9781139171762 Google Scholar: Lookup Link · Zbl 0666.13002 · doi:10.1017/CBO9781139171762 |
| [15] | S. Nasseh and S. Sather-Wagstaff, “Vanishing of Ext and Tor over fiber products”, Proc. Amer. Math. Soc. 145:11 (2017), 4661-4674. Digital Object Identifier: 10.1090/proc/13633 Google Scholar: Lookup Link · Zbl 1394.13018 · doi:10.1090/proc/13633 |
| [16] | C. Peskine and L. Szpiro, “Dimension projective finie et cohomologie locale: Applications à la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck”, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 47-119. · Zbl 0268.13008 |
| [17] | P. Roberts, “Le théorème d’intersection”, C. R. Acad. Sci. Paris Sér. I Math. 304:7 (1987), 177-180. · Zbl 0609.13010 |
| [18] | J. J. Rotman, An introduction to homological algebra, 2nd ed., Springer, New York, 2009. Digital Object Identifier: 10.1007/b98977 Google Scholar: Lookup Link · Zbl 1157.18001 · doi:10.1007/b98977 |
| [19] | P. Schenzel, “On the use of local cohomology in algebra and geometry”, pp. 241-292 in Six lectures on commutative algebra (Bellaterra, 1996), edited by J. Elías et al., Progr. Math. 166, Birkhäuser, Basel, 1998. Digital Object Identifier: 10.1007/978-3-0346-0329-4 Google Scholar: Lookup Link · Zbl 0949.13012 · doi:10.1007/978-3-0346-0329-4 |
| [20] | P. Schenzel, “On birational Macaulayfications and Cohen-Macaulay canonical modules”, J. Algebra 275:2 (2004), 751-770. Digital Object Identifier: 10.1016/j.jalgebra.2003.12.016 Google Scholar: Lookup Link · Zbl 1103.13014 · doi:10.1016/j.jalgebra.2003.12.016 |
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