Castelnuovo-Mumford regularity and projective dimension of a squarefree monomial ideal

Abstract

Let S = K[x₁, x₂, . . . , x_n] be the polynomial ring in n variables over a field K, and let I be a squarefree monomial ideal minimally generated by the monomials u₁, u₂, . . . , u_m. Let w be the smallest number t with the property that for all integers such that lcm (u_{i_1},u_{i_2},...,u_{i_t})=lcm(u₁, u₂, . . . , u_m). We give an upper bound for Castelnuovo-Mumford regularity of I by the bigsize of I. As a corollary, the projective dimension of I is bounded by the number w.

Keywords

Castelnuovo-Mumford regularity / projective dimension / squarefree monomial ideals

References

[1] Bayer D, Mumford D. What can be computed in algebraic geometry? In: Eisenbud D, Robbiano L, eds. Computational Algebraic Geometry and Commutative Algebra. Proceedings of a Conference held at Cortona, Italy, 1991. Cambridge: Cambridge Univ Press, 1993, 1–48

[2] Bayer D, Stillman M. A criterion for detecting m-regularity. Invent Math, 1987, 87: 1–11

[3] Chardin M. Some results and questions on Castelnuovo-Mumford regularity. In: Peeva I, ed. Syzygies and Hilbert Functions. Lect Notes Pure Appl Math, Vol 254. Boca Raton: CRC Press, 2007, 1–40

[4] Chardin M, Minh N C, Trung N V. On the regularity of products and intersections of complete intersections. Proc Amer Math Soc, 2007, 135(6): 1597–1606

[5] Eisenbud D. Commutative Algebra with a View Toward Algebraic Geometry.New York: Springer-Verlag, 1995

[6] Eisenbud D, Goto S. Linear free resolutions and minimal multiplicities. J Algebra, 1984, 88: 89–133

[7] Frühbis-Krüger A, Terai N. Bounds for the regularity of monomial ideals. Le Matematiche, 1998, LIII-Suppl: 83–97

[8] Herzog J. A generalization of the Taylor complex construction. Comm Algebra, 2007, 35: 1747–1756

[9] Herzog J, Hibi T. Monomial Ideals.Berlin: Springer, 2010

[10] Herzog J, Popescu D, Vladoiu M. Stanley depth and size of a monomial ideal. Proc Amer Math Soc, 2012, 140: 493–504

[11] Hoa L T, Trung N V. On the Castelnuovo-Mumford regularity and the arithmetic degree of monomial ideals. Math Z, 1998, 229: 519–537

[12] Lyubeznik G. On the arithmetical rank of monomial ideals. J Algebra, 1988, 112: 86–89

[13] Peeva I, Stillman M. Open problems on syzygies and Hilbert Functions. J Commut Algebra, 2009, 1(1): 159–195

[14] Popescu D. Stanley conjecture on intersections of four monomial prime ideals. Comm Algebra, 2003, 41: 4351–4362

[15] Stückrad J, Vogel W. Buchsbaum Rings and Applications.Berlin: Springer, 1986

[16] Sturmfels B, Trung N V, Vogel W. Bound on degrees of projective schemes. Math Ann, 1995, 302: 417–432

[17] Terai N, Hibi T. Alexander duality theorem and second Betti numbers of Stanley-Reisner rings. Adv Math, 1996, 124: 332–333

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature