Noetherian operators, primary submodules and symbolic powers

TL;DR

This paper proves the existence of Noetherian operators for primary submodules over general Noetherian rings, linking primary submodules to differential operators and introducing a new notion of differential powers that generalizes symbolic powers.

Contribution

It provides a self-contained proof of Noetherian operators' existence and introduces differential powers as a new concept generalizing symbolic powers in non-smooth settings.

Findings

Noetherian operators exist for primary submodules in broad Noetherian rings.

Differential powers coincide with symbolic powers in many non-smooth cases.

The work offers a new perspective on primary submodules via differential operators.

Abstract

We give an algebraic and self-contained proof of the existence of the so-called Noetherian operators for primary submodules over general classes of Noetherian commutative rings. The existence of Noetherian operators accounts to provide an equivalent description of primary submodules in terms of differential operators. As a consequence, we introduce a new notion of differential powers which coincides with symbolic powers in many interesting non-smooth settings, and so it could serve as a generalization of the Zariski-Nagata Theorem.