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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 quasicoherent sheaves of modules over that scheme.
Original language  English 

Article number  36 
Pages (fromto)  10771120 
Number of pages  44 
Journal  Theory and Applications of Categories 
Volume  39 
Publication status  Published  2023 
Keywords
 differential bundles
 modules
 tangent categories
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DE23: New Foundations for Algebraic Geometry
Garner, R. & Lemay, J.
10/04/23 → 9/04/26
Project: Research