Depth of powers of edge ideals of Cohen-Macaulay trees

TL;DR

This paper establishes a precise formula for the depth of powers of edge ideals of Cohen-Macaulay trees, revealing how depth decreases linearly with the power until reaching 1.

Contribution

It provides an explicit depth formula for all powers of edge ideals of Cohen-Macaulay trees, a significant extension in combinatorial commutative algebra.

Findings

Depth of $S/I^t$ decreases linearly with $t$ until reaching 1.

Depth formula holds for all $t \\ge 1$ and Cohen-Macaulay trees.

Results contribute to understanding the algebraic properties of edge ideals.

Abstract

Let $I$ be the edge ideal of a Cohen-Macaulay tree of dimension $d$ over a polynomial ring $S = \mathrm{k}[x_1,\ldots,x_{d},y_1,\ldots,y_d]$. We prove that for all $t \ge 1$, $$\operatorname{depth} (S/I^t) = \operatorname{max} \{d -t + 1, 1 \}.$$