Stability of depth and Stanley depth of symbolic powers of squarefree monomial ideals

Abstract

Let K be a field and S = K[x 1 , . . . , x n ] be the polynomial ring in n variables over K. Assume that I ⊂ S is a squarefree monomial ideal. For every integer k ≥ 1, we denote the k-th symbolic power of I by I (k) . Recently, Montaño and Núñez-Betancourt [14] proved that for every pair of integers m, k ≥ 1,). We provide an alternative proof for this inequality. Moreover, we reprove the known results that the sequence {depth(S/I (k) )} ∞ k=1 is convergent and min k depth(S/I (k) ) = lim k→∞ depth(S/I (k) ) = n − ℓ s (I),where ℓ s (I) denotes the symbolic analytic spread of I. We also determine an upper bound for the index of depth stability of symbolic powers of I. Next, we consider the Stanley depth of symbolic powers and prove that the sequences {sdepth(S/I (k) )} ∞ k=1 and {sdepth(I (k) )} ∞ k=1 are convergent and the limit of each sequence is equal to its minimum. Furthermore, we determine an upper bound for the indices of sdepth stability of symbolic powers.