Differential and symbolic powers of ideals
Alessandro De Stefani, Eloísa Grifo, Jack Jeffries
TL;DR
The work extends Zariski–Nagata-type descriptions of symbolic powers to a broad setting by using $\mathbb{Z}$-linear differential operators and, in mixed characteristic, $p$-derivations. It removes separability hypotheses and addresses nonfinitely generated modules of principal parts, establishing equal-characteristic and singular-case analogues via subfields $K_0$ with finitely generated differential structures. The main contributions include a general equal-characteristic theorem with a $K_0$-linear differential calculus for symbolic powers, a separability-free Yairon-type characterization for singular rings, and a robust mixed-characteristic framework identifying symbolic powers with mixed differential powers defined by $\delta$ and differential operators. These results provide new tools for computing symbolic powers in non-smooth and mixed-characteristic contexts and show fundamental limitations through explicit counterexamples (unramified DVRs without $p$-derivations). Overall, the paper broadens the differential-operator perspective on symbolic powers and enhances applicability to arithmetic and singular settings.
Abstract
We characterize symbolic powers of prime ideals in polynomial rings over any field in terms of $\mathbb{Z}$-linear differential operators, and of prime ideals in polynomial rings over complete discrete valuation rings with a $p$-derivation $δ$ in terms of $\mathbb{Z}$-linear differential operators and of $δ$. This extends previous work of the same authors, as it allows the removal of separability hypotheses that were otherwise necessary. The absence of separability and the fact that modules of $\mathbb{Z}$-linear differential operators are typically not finitely generated require the introduction of new techniques. As a byproduct, we extend a characterization of symbolic powers due to Cid-Ruiz which also holds in the nonsmooth case. Finally, we produce an example of an unramified discrete valuation ring that has no $p$-derivations.