Products of ideals of linear forms in quadric hypersurfaces

Conca, Aldo; Nguyen, Hop; Vu, Thanh

doi:10.1090/proc/14393

Products of ideals of linear forms in quadric hypersurfaces
HTML articles powered by AMS MathViewer

by Aldo Conca, Hop D. Nguyen and Thanh Vu

Proc. Amer. Math. Soc. 147 (2019), 1867-1880

DOI: https://doi.org/10.1090/proc/14393

Published electronically: January 18, 2019

PDF | Request permission

Abstract:

Conca and Herzog proved that any product of ideals of linear forms in a polynomial ring has a linear resolution. The goal of this paper is to establish the same result for any quadric hypersurface. The main tool we develop and use is a flexible version of Derksen and Sidman’s approximation systems.

References

J. Abbott, A.M. Bigatti and G. Lagorio, CoCoA-5: a system for doing computations in commutative algebra. Available at http://cocoa.dima.unige.it.
Luchezar L. Avramov, Aldo Conca, and Srikanth B. Iyengar, Subadditivity of syzygies of Koszul algebras, Math. Ann. 361 (2015), no. 1-2, 511–534. MR 3302628, DOI 10.1007/s00208-014-1060-4
Luchezar L. Avramov and David Eisenbud, Regularity of modules over a Koszul algebra, J. Algebra 153 (1992), no. 1, 85–90. MR 1195407, DOI 10.1016/0021-8693(92)90149-G
Luchezar L. Avramov and Irena Peeva, Finite regularity and Koszul algebras, Amer. J. Math. 123 (2001), no. 2, 275–281. MR 1828224
Giulio Caviglia, Bounds on the Castelnuovo-Mumford regularity of tensor products, Proc. Amer. Math. Soc. 135 (2007), no. 7, 1949–1957. MR 2299466, DOI 10.1090/S0002-9939-07-08222-6
M. Chardin, On the behaviour of Castelnuovo-Mumford regularity with respect to some functors. Preprint, http://arxiv.org/abs/0706.2731, (2007).
Aldo Conca, Universally Koszul algebras, Math. Ann. 317 (2000), no. 2, 329–346. MR 1764242, DOI 10.1007/s002080000100
Aldo Conca, Universally Koszul algebras defined by monomials, Rend. Sem. Mat. Univ. Padova 107 (2002), 95–99. MR 1926203
Aldo Conca, Emanuela De Negri, and Maria Evelina Rossi, Koszul algebras and regularity, Commutative algebra, Springer, New York, 2013, pp. 285–315. MR 3051376, DOI 10.1007/978-1-4614-5292-8_{8}
Aldo Conca and Jürgen Herzog, Castelnuovo-Mumford regularity of products of ideals, Collect. Math. 54 (2003), no. 2, 137–152. MR 1995137
Aldo Conca, Srikanth B. Iyengar, Hop D. Nguyen, and Tim Römer, Absolutely Koszul algebras and the Backelin-Roos property, Acta Math. Vietnam. 40 (2015), no. 3, 353–374. MR 3395644, DOI 10.1007/s40306-015-0125-0
Harm Derksen and Jessica Sidman, A sharp bound for the Castelnuovo-Mumford regularity of subspace arrangements, Adv. Math. 172 (2002), no. 2, 151–157. MR 1942401, DOI 10.1016/S0001-8708(02)00019-1
Harm Derksen and Jessica Sidman, Castelnuovo-Mumford regularity by approximation, Adv. Math. 188 (2004), no. 1, 104–123. MR 2084776, DOI 10.1016/j.aim.2003.10.001
David Eisenbud, Craig Huneke, and Bernd Ulrich, The regularity of Tor and graded Betti numbers, Amer. J. Math. 128 (2006), no. 3, 573–605. MR 2230917
D. Grayson and M. Stillman, Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2.
Jürgen Herzog and Srikanth Iyengar, Koszul modules, J. Pure Appl. Algebra 201 (2005), no. 1-3, 154–188. MR 2158753, DOI 10.1016/j.jpaa.2004.12.037
Srikanth B. Iyengar and Tim Römer, Linearity defects of modules over commutative rings, J. Algebra 322 (2009), no. 9, 3212–3237. MR 2567417, DOI 10.1016/j.jalgebra.2009.04.023
Hop D. Nguyen and Thanh Vu, Regularity over homomorphisms and a Frobenius characterization of Koszul algebras, J. Algebra 429 (2015), 103–132. MR 3320617, DOI 10.1016/j.jalgebra.2014.12.047
Hop D. Nguyen and Thanh Vu, Regularity of products over quadratic hypersurfaces. Extended Abstracts Spring 2015, Research Perspectives CRM Barcelona vol. 5, 129–133, Trends in Mathematics, Springer-Birkhäuser, Basel, 2016.
Jan-Erik Roos, Good and bad Koszul algebras and their Hochschild homology, J. Pure Appl. Algebra 201 (2005), no. 1-3, 295–327. MR 2158761, DOI 10.1016/j.jpaa.2004.12.021
Jessica Sidman, On the Castelnuovo-Mumford regularity of products of ideal sheaves, Adv. Geom. 2 (2002), no. 3, 219–229. MR 1924756, DOI 10.1515/advg.2002.010
Ngô Việt Trung, Reduction exponent and degree bound for the defining equations of graded rings, Proc. Amer. Math. Soc. 101 (1987), no. 2, 229–236. MR 902533, DOI 10.1090/S0002-9939-1987-0902533-1