Powers of sums and their associated primes

TL;DR

This paper investigates the associated primes of powers of sums of ideals in tensor product polynomial rings, providing explicit descriptions and confirming persistence properties in new cases.

Contribution

It offers a detailed analysis of associated primes of sums of ideals in tensor product rings and establishes persistence properties in broader contexts than previously known.

Findings

Associated primes of large powers of I+J are explicitly determined.

Persistence property of I+J is confirmed in many new cases.

Provides new insights into the structure of associated primes in tensor product settings.

Abstract

Let $A, B$ be polynomial rings over a field $k$, and $I\subseteq A, J\subseteq B$ proper homogeneous ideals. We analyze the associated primes of powers of $I+J\subseteq A\otimes_k B$ given the data on the summands. The associated primes of large enough powers of $I+J$ are determined. We then answer positively a question about the persistence property of $I+J$ in many new cases.