Depth and regularity of powers of sums of ideals

TL;DR

This paper studies how the algebraic invariants depth and regularity of powers of sums of ideals relate to those of the individual ideals, providing insights into their behavior and possible value sequences.

Contribution

It introduces new relationships between the depth and regularity of powers of sums of ideals and those of the original ideals, including the realization of any non-increasing sequence as a depth function.

Findings

Depth function can be any infinite non-increasing sequence of non-negative integers.

Provides formulas relating depth and regularity of $I+J$ to those of $I$ and $J$.

Enhances understanding of the asymptotic behavior of these invariants.

Abstract

Given arbitrary homogeneous ideals $I$ and $J$ in polynomial rings $A$ and $B$ over a field $k$, we investigate the depth and the Castelnuovo-Mumford regularity of powers of the sum $I+J$ in $A \otimes_k B$ in terms of those of $I$ and $J$. Our results can be used to study the behavior of the depth and regularity functions of powers of an ideal. For instance, we show that such a depth function can take as its values any infinite non-increasing sequence of non-negative integers.