Powers of sums and their homological invariants

TL;DR

This paper derives formulas for homological invariants of powers of sums of ideals in tensor product algebras, revealing how these invariants depend on the properties of the original ideals, especially in polynomial rings.

Contribution

It provides explicit formulas for homological invariants of powers of sums of ideals, utilizing Tor vanishing properties and focusing on polynomial rings over fields.

Findings

Exact formulas for depth and regularity of powers of sums of ideals.

Homological invariants depend on properties of original ideals and their Tor vanishing.

Results apply to polynomial rings over fields with characteristic zero or monomial ideals.

Abstract

Let $R$ and $S$ be standard graded algebras over a field $k$, and $I \subseteq R$ and $J \subseteq S$ homogeneous ideals. Denote by $P$ the sum of the extensions of $I$ and $J$ to $R\otimes_k S$. We investigate several important homological invariants of powers of $P$ based on the information about $I$ and $J$, with focus on finding the exact formulas for these invariants. Our investigation exploits certain Tor vanishing property of natural inclusion maps between consecutive powers of $I$ and $J$. As a consequence, we provide fairly complete information about the depth and regularity of powers of $P$ given that $R$ and $S$ are polynomial rings and either $\text{char}~ k=0$ or $I$ and $J$ are generated by monomials.