Stability of Depth and Cohen-Macaulayness of Integral Closures of Powers of Monomial Ideals
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Stability of Depth and Cohen-Macaulayness of Integral Closures of Powers of Monomial Ideals Le Tuan Hoa1 · Tran Nam Trung1
Received: 2 April 2017 / Revised: 23 June 2017 / Accepted: 11 July 2017 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2017
Abstract Let I be a monomial ideal in a polynomial ring R = k[x1 , . . . , xr ]. In this paper, we give an upper bound on dstab(I ) in terms of r and the maximal generating degree d(I ) of I such that depthR/I n is constant for all n dstab(I ). As an application, we classify the class of monomial ideals I such that I n is Cohen-Macaulay for some integer n 0. Keywords Depth · Monomial ideal · Simplicial complex · Integral closure Mathematics Subject Classification (2010) 13D45 · 05C90
1 Introduction Let R = k[x1 , . . . , xr ] be a polynomial ring over a field k and a a homogeneous ideal in R. It was shown by Brodmann [2] that depthR/an is constant for n 0. The smallest integer m > 0 such that depthR/an = depthR/am for all n m is called the index of depth stability and is denoted by dstab(a). Since the behavior of depth function depthR/an is quite mysterious (see [5, 7]), it is of great interest to bound dstab(a) in terms of r and a. However, until now this problem is only solved for a few classes of monomial ideals (see, e.g., [7, 8, 20]). The bound obtained in [20] for ideals generated by square-free monomials of degree two is rather small and optimal, and this problem is still open for a general square-free monomial ideal. In this direction, it is also of interest to consider similar problems for other powers of a. In [10] together with Kimura and Terai, we were able to solve the problem of bounding the
Le Tuan Hoa
[email protected] Tran Nam Trung [email protected] 1
Institute of Mathematics, VAST, 18 Hoang Quoc Viet, Hanoi 10307, Vietnam
L. T. Hoa, T. N. Trung
index of depth stability for symbolic powers of square-free monomial ideals. In this paper, we are interested in bounding the index of depth stability dstab(a) for integral closures, which is defined as the smallest integer m > 0 such that depthR/an = depthR/am for all n m. Like in the case of ordinary powers, dstab(a) is well-defined. We only consider the problem for monomial ideals I . In this context, one can use geometry and convex analysis to describe the integral closures of I n (see Definition 2.1 and some properties after it). Then one can use Takayama’s formula (see Lemma 2.4) to compute the local cohomology modules of R/I n . This approach was successfully applied in several papers (see, e.g., [10, 11, 19]). In particular, one can show that in the class of monomial ideals the behavior of the function depthR/I n is much better than that of depthR/I n : it is “quasi-decreasing” (see Lemma 2.5) while the function depthR/I n can be any convergent non-negative numerical function (see [5]). Our main result is Theorem 3.3, where we can give an upper bound on dstab(I ) in terms of r and the maximal generating degree d(I ) o
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