Differential bundles in commutative algebra and algebraic geometry

G. S.H. Cruttwell, Jean-Simon Pacaud Lemay

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper, we explain how the abstract notion of a differential bundle in a tangent category provides a new way of thinking about the category of modules over a commutative ring and its opposite category. MacAdam previously showed that differential bundles in the tangent category of smooth manifolds are precisely smooth vector bundles. Here we provide characterizations of differential bundles in the tangent categories of commutative rings and (affine) schemes. For commutative rings, the category of differential bundles over a commutative ring is equivalent to the category of modules over that ring. For affine schemes, the category of differential bundles over the Spec of a commutative ring is equivalent to the opposite category of modules over said ring. Finally, for schemes, the category of differential bundles over a scheme is equivalent to the opposite category of quasi-coherent sheaves of modules over that scheme.

Original languageEnglish
Article number36
Pages (from-to)1077-1120
Number of pages44
JournalTheory and Applications of Categories
Volume39
Publication statusPublished - 2023

Keywords

  • differential bundles
  • modules
  • tangent categories

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