Linearly presented modules and bounds on the Castelnuovo-Mumford regularity of ideals

Caviglia, Giulio; De Stefani, Alessandro

doi:10.1090/proc/15902

Linearly presented modules and bounds on the Castelnuovo-Mumford regularity of ideals
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Proc. Amer. Math. Soc. 150 (2022), 1397-1404 Request permission

Abstract:

We estimate the Castelnuovo-Mumford regularity of ideals in a polynomial ring over a field by studying the regularity of certain modules generated in degree zero and with linear relations. In dimension one, this process gives a new type of upper bounds. By means of recursive techniques this also produces new upper bounds for ideals in any dimension.

References

Winfried Bruns, Aldo Conca, and Tim Römer, Koszul cycles, Combinatorial aspects of commutative algebra and algebraic geometry, Abel Symp., vol. 6, Springer, Berlin, 2011, pp. 17–33. MR 2810423, DOI 10.1007/978-3-642-19492-4_{2}
M. Brodmann and T. Götsch, Bounds for the Castelnuovo-Mumford regularity, J. Commut. Algebra 1 (2009), no. 2, 197–225. MR 2504932, DOI 10.1216/JCA-2009-1-2-197
Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
Dave Bayer and David Mumford, What can be computed in algebraic geometry?, Computational algebraic geometry and commutative algebra (Cortona, 1991) Sympos. Math., XXXIV, Cambridge Univ. Press, Cambridge, 1993, pp. 1–48. MR 1253986
David Bayer and Michael Stillman, On the complexity of computing syzygies, J. Symbolic Comput. 6 (1988), no. 2-3, 135–147. Computational aspects of commutative algebra. MR 988409, DOI 10.1016/S0747-7171(88)80039-7
Giulio Caviglia and Alessandro De Stefani, The Eisenbud-Green-Harris conjecture for fast-growing degree sequences, arXiv:2007.15467, 2020.
Marc Chardin, Amadou Lamine Fall, and Uwe Nagel, Bounds for the Castelnuovo-Mumford regularity of modules, Math. Z. 258 (2008), no. 1, 69–80. MR 2350034, DOI 10.1007/s00209-007-0157-9
Giulio Caviglia and Diane Maclagan, Some cases of the Eisenbud-Green-Harris conjecture, Math. Res. Lett. 15 (2008), no. 3, 427–433. MR 2407220, DOI 10.4310/MRL.2008.v15.n3.a3
Giulio Caviglia and Enrico Sbarra, Characteristic-free bounds for the Castelnuovo-Mumford regularity, Compos. Math. 141 (2005), no. 6, 1365–1373. MR 2188440, DOI 10.1112/S0010437X05001600
David Eisenbud and Shiro Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), no. 1, 89–133. MR 741934, DOI 10.1016/0021-8693(84)90092-9
David Eisenbud, Mark Green, and Joe Harris, Higher Castelnuovo theory, Astérisque 218 (1993), 187–202. Journées de Géométrie Algébrique d’Orsay (Orsay, 1992). MR 1265314
David Eisenbud, Mark Green, and Joe Harris, Cayley-Bacharach theorems and conjectures, Bull. Amer. Math. Soc. (N.S.) 33 (1996), no. 3, 295–324. MR 1376653, DOI 10.1090/S0273-0979-96-00666-0
André Galligo, À propos du théorème de-préparation de Weierstrass, Fonctions de plusieurs variables complexes (Sém. François Norguet, octobre 1970–décembre 1973; à la mémoire d’André Martineau), Lecture Notes in Math., Vol. 409, Springer, Berlin, 1974, pp. 543–579 (French). Thèse de 3ème cycle soutenue le 16 mai 1973 à l’Institut de Mathématique et Sciences Physiques de l’Université de Nice. MR 0402102
André Galligo, Théorème de division et stabilité en géométrie analytique locale, Ann. Inst. Fourier (Grenoble) 29 (1979), no. 2, vii, 107–184 (French, with English summary). MR 539695
M. Giusti, Some effectivity problems in polynomial ideal theory, EUROSAM 84 (Cambridge, 1984) Lecture Notes in Comput. Sci., vol. 174, Springer, Berlin, 1984, pp. 159–171. MR 779123, DOI 10.1007/BFb0032839
Mark Green, Restrictions of linear series to hyperplanes, and some results of Macaulay and Gotzmann, Algebraic curves and projective geometry (Trento, 1988) Lecture Notes in Math., vol. 1389, Springer, Berlin, 1989, pp. 76–86. MR 1023391, DOI 10.1007/BFb0085925
Lê Tuân Hoa and Eero Hyry, Castelnuovo-Mumford regularity of canonical and deficiency modules, J. Algebra 305 (2006), no. 2, 877–900. MR 2266858, DOI 10.1016/j.jalgebra.2006.05.001
Jee Koh, Ideals generated by quadrics exhibiting double exponential degrees, J. Algebra 200 (1998), no. 1, 225–245. MR 1603272, DOI 10.1006/jabr.1997.7225
Ernst W. Mayr and Albert R. Meyer, The complexity of the word problems for commutative semigroups and polynomial ideals, Adv. in Math. 46 (1982), no. 3, 305–329. MR 683204, DOI 10.1016/0001-8708(82)90048-2