Tensor categorical foundations of algebraic geometry

TL;DR

This paper develops a tensor categorical framework to generalize and globalize classical algebraic geometry constructions, connecting schemes and stacks with their tensor categories of quasi-coherent sheaves.

Contribution

It introduces a foundational approach to algebraic geometry via cocomplete tensor categories, extending local-global principles and universal properties to a categorical setting.

Findings

Tensor categorical globalizations of classical morphisms

Development of commutative algebra in cocomplete tensor categories

Universal properties of moduli spaces translated into tensor categories

Abstract

Tannaka duality and its extensions by Lurie, Sch\"appi et al. reveal that many schemes as well as algebraic stacks may be identified with their tensor categories of quasi-coherent sheaves. In this thesis we study constructions of cocomplete tensor categories (resp. cocontinuous tensor functors) which usually correspond to constructions of schemes (resp. their morphisms) in the case of quasi-coherent sheaves. This means to globalize the usual local-global algebraic geometry. For this we first have to develop basic commutative algebra in an arbitrary cocomplete tensor category. We then discuss tensor categorical globalizations of affine morphisms, projective morphisms, immersions, classical projective embeddings (Segre, Pl\"ucker, Veronese), blow-ups, fiber products, classifying stacks and finally tangent bundles. It turns out that the universal properties of several moduli spaces or stacks translate to the corresponding tensor categories.