Differential Bundles in Commutative Algebra and Algebraic Geometry

TL;DR

This paper explores how differential bundles in tangent categories relate to modules and sheaves in algebraic geometry, providing new categorical characterizations for rings, schemes, and vector bundles.

Contribution

It introduces a categorical framework connecting differential bundles with modules and sheaves in algebraic geometry, extending previous work on smooth manifolds.

Findings

Differential bundles over a commutative ring are equivalent to modules.

Differential bundles over affine schemes correspond to the opposite category of modules.

Over schemes, differential bundles are equivalent to the opposite category of quasi-coherent sheaves.

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.