Depths of Powers of the Edge Ideal of a Tree

Abstract

Lower bounds are given for the depths of R/I ^t for t ≥ 1 when I is the edge ideal of a tree or forest. The bounds are given in terms of the diameter of the tree, or in case of a forest, the largest diameter of a connected component and the number of connected components. These lower bounds provide a lower bound on the power for which the depths stabilize.

Key Words:

• Depth
• Edge ideal
• Monomial ideal
• Powers of ideals
• Tree

2000 Mathematics Subject Classification:

• 13A17
• 13F55
• 05C65
• 90C27

ACKNOWLEDGMENTS

The author would like to thank Jürgen Herzog for asking the question that sparked my initial interest in the depths of powers of edge ideals. The author would also like to thank Nate Dean for useful conversations regarding graph theory and Ray Heitmann and Rafael Villarreal for helpful suggestions regarding an early version of the manuscript.

Notes

Communicated by R. Villarreal.