Let \mathbbK be a field and S=\mathbbK[x_{1},...,x_{n}] be the
polynomial ring in n variables over the field \mathbbK. Let G be a
forest with p connected components G_{1},…,G_{p} and let I=I(G) be its
edge ideal in S. Suppose that d_{i} is the diameter of G_{i}, 1 ≤ i ≤ p, and consider d = max{d_{i}  1 ≤ i ≤ p}.
Morey has shown that for every t ≥ 1, the quantity max{⎡[(d−t+2)/3]⎤+p−1,p} is a lower bound for depth(S/I^{t}). In this paper, we show that for every t ≥ 1, the
mentioned quantity is also a lower bound for 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.
Download TeX format
