Stanley depth of powers of the edge ideal of a forest

Abstract

Let K \mathbb {K} be a field and S = K [ x 1 , … , x n ] S=\mathbb {K}[x_1,\dots ,x_n] be the polynomial ring in n n variables over the field K \mathbb {K} . Let G G be a forest with p p connected components G 1 , … , G p G_1,\ldots ,G_p and let I = I ( G ) I=I(G) be its edge ideal in S S . Suppose that d i d_i is the diameter of G i G_i , 1 ≤ i ≤ p 1\leq i\leq p , and consider d = max { d i ∣ 1 ≤ i ≤ p } d =\max \hspace {0.04cm}\{d_i\mid 1\leq i\leq p\} . Morey has shown that for every t ≥ 1 t\geq 1 , the quantity max { ⌈ d − t + 2 3 ⌉ + p − 1 , p } \max \{\lceil \frac {d-t+2}{3}\rceil +p-1,p\} is a lower bound for depth ( S / I t ) \textrm {depth}(S/I^t) . In this paper, we show that for every t ≥ 1 t\geq 1 , the mentioned quantity is also a lower bound for sdepth ( S / I t ) \textrm {sdepth}(S/I^t) . By combining this inequality with Burch’s inequality, we show that any sufficiently large powers of edge ideals of forests are Stanley. Finally, we state and prove a generalization of our main theorem.