Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Commutative algebra in tensor categories

Kevin Coulembier

Abstract

We develop some foundations of commutative algebra, with a view towards algebraic geometry, in symmetric tensor categories. Most results establish analogues of classical theorems, in tensor categories which admit a tensor functor to some tensor category verifying specific conditions. This is in line with the current program which aims to describe tensor categories by their tensor functors to incompressible categories. We place particular emphasis on the notion of observable subgroups of affine group schemes in tensor categories, which in particular leads to some further insight into observability for classical affine group schemes.

Commutative algebra in tensor categories

Abstract

We develop some foundations of commutative algebra, with a view towards algebraic geometry, in symmetric tensor categories. Most results establish analogues of classical theorems, in tensor categories which admit a tensor functor to some tensor category verifying specific conditions. This is in line with the current program which aims to describe tensor categories by their tensor functors to incompressible categories. We place particular emphasis on the notion of observable subgroups of affine group schemes in tensor categories, which in particular leads to some further insight into observability for classical affine group schemes.
Paper Structure (45 sections, 67 theorems, 100 equations)

This paper contains 45 sections, 67 theorems, 100 equations.

Table of Contents

  1. Preliminaries
  2. Tensor categories
  3. Affine group schemes
  4. Infinitesimal group schemes
  5. Additive group
  6. The right adjoint of a tensor functor
  7. Higher Verlinde categories
  8. The basics of commutative algebra
  9. Algebras
  10. Ideals
  11. Prime ideals
  12. Radical ideals
  13. Nil ideals
  14. Algebra morphisms
  15. Grothendieck topologies
  16. ...and 30 more sections

Key Result

Proposition 1

The following conditions are equivalent on a tensor functor $F:\mathcal{C}\to\mathcal{D}$ between pretannakian categories:

Theorems & Definitions (163)

  • Example 1.2.4
  • Proposition 1
  • proof
  • Remark 2
  • Lemma 3
  • proof
  • Corollary 2.2.4
  • Corollary 2.2.5
  • proof
  • Corollary 2.2.6
  • ...and 153 more