When does depth stabilize early on?

2014/07/15 by Nam, Le Dinh, Varbaro, Matteo
#13A15 #13A30 #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1407.3967

Abstract

In this paper we study graded ideals I in a polynomial ring S such that the numerical function f(k)=depth(S/I^k) is constant. We show that, if (i) the Rees algebra of I is Cohen-Macaulay, (ii) the cohomological dimension of I is not larger than the projective dimension of S/I and (iii) the K-algebra generated by some generators of I is a direct summand of S, then f(k) is constant. When I is a square-free monomial ideal, the above criterion includes as special cases all the results of a recent paper by Herzog and Vladoiu. In this combinatorial setting there is a chance that the converse of the above fact holds true.

Related

• Depth of an initial ideal
• An Increasing normalized depth function
• Depth of some special monomial ideals
• The depth of the associated graded ring of ideals with any reduction number
• A general formula for the index of depth stability of edge ideals