Regularity of Sn-invariant monomial ideals

For a polynomial ring S in n variables, we consider the natural action of the symmetric group S_n on S by permuting the variables. For an S_n-invariant monomial ideal I in S and j >= 0, we give an explicit recipe for computing the modules Ext^j(S/I,S), and use this to describe the projective dimension and regularity of I. We classify the S_n-invariant monomial ideals that have a linear free resolution, and also characterize those which are Cohen-Macaulay. We then consider two settings for analyzing the asymptotic behavior of regularity: one where we look at powers of a fixed ideal I, and another where we vary the dimension of the ambient polynomial ring and examine the invariant monomial ideals induced by I. In the first case we determine the asymptotic regularity for those ideals I that are generated by the S_n-orbit of a single monomial by solving an integer linear optimization problem. In the second case we describe the behavior of regularity for any I, recovering a recent result of Murai.

[1]  On the ideal generated by all squarefree monomials of a given degree , 2016, Journal of Commutative Algebra.

[2]  Claudiu Raicu.  Regularity and cohomology of determinantal thickenings , 2016, 1611.00415.

[3]  P. Hall.  On Representatives of Subsets , 1935 .

[4]  Michael Perlman,et al.  Regularity and cohomology of Pfaffian thickenings , 2017, Journal of Commutative Algebra.

[5]  Aldo Conca,et al.  Castelnuovo-Mumford regularity of products of ideals , 2002 .

[6]  Seth Sullivant,et al.  Finite Groebner bases in infinite dimensional polynomial rings and applications , 2009, 0908.1777.

[7]  A. Banerjee.  The regularity of powers of edge ideals , 2014, 1406.0456.

[8]  David Eisenbud,et al.  Cohomology on Toric Varieties and Local Cohomology with Monomial Supports , 2000, J. Symb. Comput..

[9]  Matthias Aschenbrenner,et al.  Finite generation of symmetric ideals , 2004, math/0411514.

[10]  B. Sturmfels,et al.  Combinatorial Commutative Algebra , 2004 .

[11]  Vijay Kodiyalam.  Asymptotic behaviour of Castelnuovo-Mumford regularity , 1999 .

[12]  Sakinah,et al.  Vol. , 2020, New Medit.

[13]  J. Herzog,et al.  Asymptotic Behaviour of the Castelnuovo-Mumford Regularity , 1999, Compositio Mathematica.

[14]  Local cohomology at monomial ideals , 2000, math/0001153.

[15]  Uwe Nagel,et al.  Codimension and projective dimension up to symmetry , 2018, Mathematische Nachrichten.

[16]  Betti tables of monomial ideals fixed by permutations of the variables , 2019, 1907.09727.

[17]  D. Eisenbud,et al.  Young diagrams and determinantal varieties , 1980 .

[18]  U. Nagel,et al.  Betti numbers of symmetric shifted ideals , 2019, 1907.04288.

[19]  Huy Tài Hà,et al.  Regularity of powers of edge ideals , 2014 .

[20]  Eran Nevo,et al.  C4-free edge ideals , 2013 .

[21]  Moritz Beckmann,et al.  Young tableaux , 2007 .

[22]  Kohji Yanagawa.  Alexander Duality for Stanley–Reisner Rings and Squarefree Nn-Graded Modules , 2000 .

[23]  Uwe Nagel,et al.  Castelnuovo–Mumford Regularity up to Symmetry , 2018, 1806.00457.

[24]  D. E Cohen,et al.  On the laws of a metabelian variety , 1967 .

[25]  Bernd Sturmfels,et al.  Cellular resolutions of monomial modules , 1997 .

[26]  R. Stanley.  Combinatorics and commutative algebra , 1983 .

[27]  J. Weyman,et al.  Local cohomology with support in generic determinantal ideals , 2013, 1309.0617.