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Torsion modules and differential operators in infinitely many variables

Leonid Positselski

TL;DR

This work generalizes Grothendieck’s differential operators to infinitely many variables by recasting them as I-torsion elements in the bimodule of all operators, relative to the diagonal ideal in $T=R\otimes_KR$. It introduces three torsion notions (s-torsion, strongly $I$-torsion, and quite $I$-torsion) and three corresponding classes of quasi-modules, along with ordinal-valued differential operators that realize transfinite orders. The paper proves that every ordinal can be realized as the order of a differential operator on polynomial rings in infinitely many variables, and it develops a robust theory of localization and colocalization of differential operators under flat epimorphisms, including extensions to both quasi- and contraherent-cosheaf contexts. In addition, it establishes a precise framework for sheaf- and cosheaf-like constructions of differential operators on schemes, and provides sharp bounds and realizations for the ordinal lengths of operator orders. The results illuminate the structure of differential operators beyond finite order, with potential geometric applications in D-module theory and the study of infinite-variable algebras.

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.

Torsion modules and differential operators in infinitely many variables

TL;DR

This work generalizes Grothendieck’s differential operators to infinitely many variables by recasting them as I-torsion elements in the bimodule of all operators, relative to the diagonal ideal in . It introduces three torsion notions (s-torsion, strongly -torsion, and quite -torsion) and three corresponding classes of quasi-modules, along with ordinal-valued differential operators that realize transfinite orders. The paper proves that every ordinal can be realized as the order of a differential operator on polynomial rings in infinitely many variables, and it develops a robust theory of localization and colocalization of differential operators under flat epimorphisms, including extensions to both quasi- and contraherent-cosheaf contexts. In addition, it establishes a precise framework for sheaf- and cosheaf-like constructions of differential operators on schemes, and provides sharp bounds and realizations for the ordinal lengths of operator orders. The results illuminate the structure of differential operators beyond finite order, with potential geometric applications in D-module theory and the study of infinite-variable algebras.

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.
Paper Structure (23 sections, 49 theorems, 65 equations)

This paper contains 23 sections, 49 theorems, 65 equations.

Table of Contents

  1. Three Classes of Torsion Modules
  2. Torsion modules
  3. Strongly torsion modules
  4. Quite torsion modules
  5. Three Classes of Quasi-Modules
  6. Quasi-modules
  7. Strong quasi-modules
  8. Quite quasi-modules
  9. Three Classes of Differential Operators
  10. Differential operators of finite order
  11. Differential operators of transfinite order
  12. Differential operators having no order
  13. Bounding and Realizing Ordinals
  14. Flat Epimorphisms of Commutative Rings and Quasi-Modules
  15. Flat epimorphisms and quite quasi-modules
  16. ...and 8 more sections

Key Result

Lemma 1.1

The class of $s$-torsion $T$-modules $T{\operatorname{\mathsf{--Mod}}}_{s{\operatorname{\mathsf{-tors}}}}$ is closed under subobjects, quotients, extensions, and infinite direct sums in $T{\operatorname{\mathsf{--Mod}}}$.

Theorems & Definitions (107)

  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • proof
  • Lemma 1.5
  • proof
  • ...and 97 more