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Regularity bounds for complexes and their homology

  • HOP D. NGUYEN (a1) (a2)

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.

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[1]Avramov, L.L. and Eisenbud, D.Regularity of modules over a Koszul algebra. J. Algebra. 153 (1992), 8590.
[2]Avramov, L.L. and Foxby, H.-B.Ring homomorphisms and finite Gorenstein dimension. Proc. London Math. Soc. 75 (1997), 241270.
[3]Avramov, L.L., Iyengar, S.B. and Miller, C.Homology over local homomorphisms. Amer. J. Math. 128 (2006), no. 1, 2390.
[4]Avramov, L.L. and Peeva, I.Finite regularity and Koszul algebras. Amer. J. Math. 123 (2001), 275281.
[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. 148.
[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. 285315.
[7]Bruns, W. and Herzog, J.Cohen–Macaulay rings. Rev. ed. Cambridge Studies in Advanced Math. 39 (Cambridge University Press, 1998).
[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), 3948.
[9]Caviglia, G.Bounds on the Castelnuovo–Mumford regularity of tensor products. Proc. Amer. Math. Soc. 135, no. 7 (2007), 19491957.
[10]Chardin, M. On the behaviour of Castelnuovo–Mumford regularity with respect to some functors. Preprint (2007), http://arxiv.org/abs/0706.2731.
[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. 140.
[12]Chardin, M. and Divaani-Aazar, K.Generalised local cohomology and regularity of Ext modules. J. Algebra 319 (2008), 47804797.
[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), 34393456.
[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.
[15]Conca, A.Koszul algebras and their syzygies. In: Combinatorial Algebraic Geometry. Conca, A.et al., Lecture Notes in Math. 2108 (Springer, 2014), pp. 131.
[16]Conca, A. and Herzog, J.Castelnuovo–Mumford regularity of products of ideals. Collect. Math. 54 (2003), 137152.
[17]Cutkosky, S.D., Herzog, J. and Trung, N.V.Asymptotic behaviour of the Castelnuovo–Mumford regularity. Compositio Math. 118, no. 3 (1999), 243261.
[18]Eisenbud, D. and Goto, S.Linear free resolutions and minimal multiplicities. J. Algebra. 88 (1984), 107184.
[19]Eisenbud, D., Huneke, C. and Ulrich, B.The regularity of Tor and graded Betti numbers. Amer. J. Math. 128, no. 3 (2006), 573605.
[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. 119137.
[21]Iyengar, S.B.Depth for complexes, and intersection theorems. Math. Z. 230 (1999), 545567.
[22]Jørgensen, P.Non-commutative Castelnuovo–Mumford regularity. Math. Proc. Camb. Phil. Soc. 125 (1999), 203221.
[23]Jørgensen, P.Linear free resolutions over non-commutative algebras. Compositio Math. 140 (2004), 10531058.
[24]Kodiyalam, V.Asymptotic behaviour of Castelnuovo–Mumford regularity. Proc. Amer. Math. Soc. 128 (2000), no. 2, 407411.
[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. 3989.
[26]Nguyen, H.D. and Vu, T.Regularity over homomorphisms and a Frobenius characterisation of Koszul algebras. J. Algebra 429 (2015), 103132.
[27]Peeva, I. and Stillman, M.Open problems on syzygies and Hilbert functions. J. Commut. Algebra. 1, no. 1 (2009), 159195.
[28]Römer, T.On the regularity over positively graded algebras. J. Algebra 319 (2008), 115.
[29]Trung, N.V.Reduction exponent and degree bound for the defining equations of graded rings. Proc. Amer. Math. Soc. 101 (2) (1987), 229236.
[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), 44334453.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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