Torsion modules and differential operators in infinitely many variables

TL;DR

This paper explores the theory of differential operators in infinitely many variables through the lens of torsion modules, introducing new concepts like transfinite orders and local orders, with numerous examples including ordinal realizations.

Contribution

It develops a framework connecting torsion modules and differential operators in infinite variable settings, including novel notions like transfinite and local orders.

Findings

Every ordinal can be realized as the order of a differential operator.

Differential operators of transfinite and local orders are constructed and analyzed.

Extensions of differential operators to localizations and colocalizations are discussed.

Abstract

This paper grew out of the author's work on arXiv:2504.18460. Differential operators in the sense of Grothendieck acting between modules over a commutative ring can be interpreted as torsion elements in the bimodule of all operators with respect to the diagonal ideal in the tensor square of the ring. Various notions of torsion modules for an infinitely generated ideal in a commutative ring lead to various notions of differential operators. We discuss differential operators of transfinite orders and differential operators having no global order at all, but only local orders with respect to specific elements of the ring. Many examples are presented. In particular, we prove that every ordinal can be realized as the order of a differential operator acting on the algebra of polynomials in infinitely many variables over a field. We also discuss extension of differential operators to localizations of rings and modules, and to colocalizations of modules.