Betti splittings for powers of sums of ideals

TL;DR

This paper investigates the algebraic properties of powers of sums of ideals in polynomial rings, providing formulas and asymptotic behavior for depth and regularity, and establishing Betti splittings under certain conditions.

Contribution

It offers new exact formulas and asymptotic descriptions for the depth and regularity of powers of sums of ideals, extending previous research and introducing Betti splittings for these ideals.

Findings

Derived formulas for depth and regularity of powers of sums of ideals

Established Betti splittings for all powers of sums of ideals under certain conditions

Extended previous work by H ext{.}T. H ext{.}a, N ext{.}V. Trung, and T ext{.}N. Trung

Abstract

Let $A$ and $B$ be standard graded polynomial rings over a field $k$ and $I$ and $J$ be non-zero, proper homogeneous ideals contained in $A$ and $B$, respectively. Denote by $P$ the sum of $I$ and $J$ in $R=A\otimes_k B$. Under reasonable conditions on $k, I$ and $J$, we provide exact formulas and describe the asymptotic behavior of the depth and the regularity of the powers of $P$ in terms of the data of $I$ and $J$. Thereby, we strengthen previous work of H.T. H\`a, N.V. Trung and T.N. Trung. Our main technical result says that, under the aforementioned conditions, for all $s\ge 0$ and all $n\ge 1$, the simple decomposition $I^sP^n=I^{s+1}P^{n-1}+I^sJ^n$ yields a Betti splitting for $I^sP^n$. A decomposition of an ideal $L$ as a sum of two subideals is called a Betti splitting if the minimal free resolution of $L$ is completely determined by those of the summands and their intersection.