# Symbolic powers of sums of ideals

## TL;DR

This paper establishes a formula for symbolic powers of sums of ideals in Noetherian algebras and provides explicit depth and regularity formulas in polynomial rings, advancing understanding of their algebraic properties.

## Contribution

The paper proves a new expansion formula for symbolic powers of sums of ideals and derives explicit formulas for depth and regularity in specific cases, with novel insights into their algebraic structure.

## Key findings

- Symbolic powers of sums decompose as sums of products of individual symbolic powers.
- Explicit depth and regularity formulas are obtained for polynomial rings in characteristic zero or for monomial ideals.
- The induced Tor maps vanish under certain conditions, revealing new structural properties.

## Abstract

Let $I$ and $J$ be nonzero ideals in two Noetherian algebras $A$ and $B$ over a field $k$. Let $I+J$ denote the ideal generated by $I$ and $J$ in $A\otimes_k B$. We prove the following expansion for the symbolic powers: $$(I+J)^{(n)} = \sum_{i+j = n} I^{(i)} J^{(j)}.$$ If $A$ and $B$ are polynomial rings and if chara$(k) = 0$ or if $I$ and $J$ are monomial ideals, we give exact formulas for the depth and the Castelnuovo-Mumford regularity of $(I+J)^{(n)}$, which depend on the interplay between the symbolic powers of $I$ and $J$. The proof involves a result of independent interest which states that under the above assumption, the induced map Tor$_i^A(k,I^{(n)}) \to$ Tor$_i^A(k,I^{(n-1)})$ is zero for all $i \ge 0$, $n \ge 0$. We also investigate other properties and invariants of $(I+J)^{(n)}$ such as the equality between ordinary and symbolic powers, the Waldschmidt constant and the Cohen-Macaulayness.