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Regularity bounds for complexes and their homology

Published online by Cambridge University Press:  02 July 2015

HOP D. NGUYEN
HOP D. NGUYEN*
Affiliation:
Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146 Genoa, Italy. Fachbereich Mathematik/Informatik, Institut für Mathematik, Universität Osnabrück, Albrectstr. 28a, 49069 Osnabrück, Germany. e-mail: [email protected]
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Abstract

Let R be a standard graded algebra over a field k. We prove an Auslander–Buchsbaum formula for the absolute Castelnuovo–Mumford regularity, extending important cases of previous works of Chardin and Römer. For a bounded complex of finitely generated graded R-modules L, we prove the equality reg L = maxi ∈$_{\mathbb Z}$ {reg Hi(L) − i} given the condition depth Hi(L) ⩾ dim Hi+1(L) - 1 for all i < sup L. As applications, we recover previous bounds on regularity of Tor due to Caviglia, Eisenbud–Huneke–Ulrich, among others. We also obtain strengthened results on regularity bounds for Ext and for the quotient by a linear form of a module.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 159 , Issue 2 , September 2015 , pp. 355 - 377
Copyright
Copyright © Cambridge Philosophical Society 2015 

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References

REFERENCES

[1] Avramov, L.L. and Eisenbud, D. Regularity of modules over a Koszul algebra. J. Algebra. 153 (1992), 85–90.Google Scholar
[2] Avramov, L.L. and Foxby, H.-B. Ring homomorphisms and finite Gorenstein dimension. Proc. London Math. Soc. 75 (1997), 241–270.Google Scholar
[3] Avramov, L.L., Iyengar, S.B. and Miller, C. Homology over local homomorphisms. Amer. J. Math. 128 (2006), no. 1, 23–90.Google Scholar
[4] Avramov, L.L. and Peeva, I. Finite regularity and Koszul algebras. Amer. J. Math. 123 (2001), 275–281.Google Scholar
[5] Bayer, D. and Mumford, D. What can be computed in algebraic geometry? In: Computational Algebraic Geometry and Commutative Algebra, Proceedings. Cortona 1991, Eisenbud, D. and Robbiano, L. (eds). (Cambridge University Press, 1993), pp. 1–48.Google Scholar
[6] Brodmann, M., Linh, C.H. and Seiler, M.-H. Castelnuovo–Mumford regularity of annihilators, Ext and Tor modules. In: Commutative Algebra: expository papers dedicated to David Eisenbud on the occasion of his 65th birthday. Peeva, I. (ed.) (Springer, 2013), pp. 285–315.Google Scholar
[7] Bruns, W. and Herzog, J. Cohen–Macaulay rings. Rev. ed. Cambridge Studies in Advanced Math. 39 (Cambridge University Press, 1998).Google Scholar
[8] Catalano-Johnson, M.L. The resolution of the ideal of 2 × 2 minors of a 2 × n matrix of linear forms. J. Algebra. 187 (1997), 39–48.Google Scholar
[9] Caviglia, G. Bounds on the Castelnuovo–Mumford regularity of tensor products. Proc. Amer. Math. Soc. 135, no. 7 (2007), 1949–1957.Google Scholar
[10] Chardin, M. On the behaviour of Castelnuovo–Mumford regularity with respect to some functors. Preprint (2007), http://arxiv.org/abs/0706.2731.Google Scholar
[11] Chardin, M. Some results and questions on Castelnuovo–Mumford regularity. In: Syzygies and Hilbert Functions. Lecture Notes in Pure and Appl. Math. 254 (2007), pp. 1–40.Google Scholar
[12] Chardin, M. and Divaani-Aazar, K. Generalised local cohomology and regularity of Ext modules. J. Algebra 319 (2008), 4780–4797.Google Scholar
[13] Chardin, M., Ha, D.T. and Hoa, L.T. Castelnuovo–Mumford regularity of Ext modules and homological degree. Trans. Amer. Math. Soc. 363 no. 7 (2011), 3439–3456.Google Scholar
[14] Christensen, L.W. and Foxby, H.-B. Hyperhomological algebra with applications to commutative rings. Preprint (2006), available online at the following address: http://www.math.ttu.edu/~lchriste/download/918-final.pdf.Google Scholar
[15] Conca, A. Koszul algebras and their syzygies. In: Combinatorial Algebraic Geometry. Conca, A. et al., Lecture Notes in Math. 2108 (Springer, 2014), pp. 1–31.Google Scholar
[16] Conca, A. and Herzog, J. Castelnuovo–Mumford regularity of products of ideals. Collect. Math. 54 (2003), 137–152.Google Scholar
[17] Cutkosky, S.D., Herzog, J. and Trung, N.V. Asymptotic behaviour of the Castelnuovo–Mumford regularity. Compositio Math. 118, no. 3 (1999), 243–261.Google Scholar
[18] Eisenbud, D. and Goto, S. Linear free resolutions and minimal multiplicities. J. Algebra. 88 (1984), 107–184.Google Scholar
[19] Eisenbud, D., Huneke, C. and Ulrich, B. The regularity of Tor and graded Betti numbers. Amer. J. Math. 128, no. 3 (2006), 573–605.Google Scholar
[20] Foxby, H.-B. and Iyengar, S.B. Depth and amplitude for unbounded complexes. In: Commutative Algebra (Grenoble/Lyon, 2001) Contemp. Math. 331 (Amer. Math. Soc., Providence, RI, 2003), pp. 119–137.Google Scholar
[21] Iyengar, S.B. Depth for complexes, and intersection theorems. Math. Z. 230 (1999), 545–567.Google Scholar
[22] Jørgensen, P. Non-commutative Castelnuovo–Mumford regularity. Math. Proc. Camb. Phil. Soc. 125 (1999), 203–221.Google Scholar
[23] Jørgensen, P. Linear free resolutions over non-commutative algebras. Compositio Math. 140 (2004), 1053–1058.Google Scholar
[24] Kodiyalam, V. Asymptotic behaviour of Castelnuovo–Mumford regularity. Proc. Amer. Math. Soc. 128 (2000), no. 2, 407–411.Google Scholar
[25] Lipman, J. Lectures on local cohomology and duality. In: Local cohomology and its applications (Guanajuoto, Mexico). Lecture Notes Pure Appl. Math. 226 (Marcel Dekker, 2002), pp. 39–89.Google Scholar
[26] Nguyen, H.D. and Vu, T. Regularity over homomorphisms and a Frobenius characterisation of Koszul algebras. J. Algebra 429 (2015), 103–132.Google Scholar
[27] Peeva, I. and Stillman, M. Open problems on syzygies and Hilbert functions. J. Commut. Algebra. 1, no. 1 (2009), 159–195.Google Scholar
[28] Römer, T. On the regularity over positively graded algebras. J. Algebra 319 (2008), 1–15.Google Scholar
[29] Trung, N.V. Reduction exponent and degree bound for the defining equations of graded rings. Proc. Amer. Math. Soc. 101 (2) (1987), 229–236.Google Scholar
[30] Zaare-Nahandi, R. and Zaare–Nahandi, R. Gröbner basis and free resolution of the ideal of 2-minors of a 2 × n matrix of linear forms. Comm. Algebra 28 (2000), 4433–4453.Google Scholar