Symbolic powers via extension
Sankhaneel Bisui, Haoxi Hu
TL;DR
The paper investigates when symbolic powers commute with extension under ring maps and extends resurgence results to broader settings. It proves a flat-extension criterion: if $\ ext{Ass}^*(IB)=\{\mathfrak p B \\mid \\mathfrak p\in \text{Ass}^*(I)\}$ and each associated prime of $I$ contracts from $B$, then $(IB)^{(n)}=I^{(n)}B$ for all $n\ge1$, with consequences that $\rho(IB)=\rho(I)$ and $\rho_a(IB)=\rho_a(I)$. Building on this, the authors extend resurgence results for sums of ideals from polynomial rings to sums $I+J$ in $R=A\otimes_{\mathbbm k} B$ where $A,B$ are finitely generated $\mathbbm k$-algebra domains over an algebraically closed field $\mathbbm k$, establishing that $\rho_a(I+J)=\max\{\rho_a(I),\rho_a(J)\}$ (and equal to $\max\{\rho_a(IR),\rho_a(JR)\}$) and providing sharp numerical bounds for $\rho(I+J)$. The work broadens the applicability of symbolic-power containment and resurgence analyses beyond polynomial rings to more general flat extensions and tensor-product domains, with geometric interpretations via fibers and prime-extension properties.
Abstract
This article investigates under which conditions the symbolic powers of the extension of an ideal is the same as the extension of the symbolic powers. Our result generalizes the known scenarios. As an application, we prove formulas for the resurgence of sum of two homogeneous ideals in finitely generated k-algebra domains, where k is algebraically closed. Initially, these were known for ideals in polynomial rings.