Symbolic powers of sums of ideals

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:

$$\begin{aligned} (I+J)^{(n)} = \sum _{i+j = n} I^{(i)} J^{(j)}. \end{aligned}$$

If A and B are polynomial rings and if ${{\,\mathrm{char}\,}}(k) = 0$ 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 the induced map ${{\,\mathrm{Tor}\,}}_i^A(k,I^{(n)}) \rightarrow {{\,\mathrm{Tor}\,}}_i^R(k,I^{(n-1)})$ is zero for any homogeneous ideal I and $i \ge 0$, $n \ge 0$. 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.