Affine functors and duality

TL;DR

This paper introduces affine functors, showing they are limits of affine schemes, and explores their structures and dualities, with applications to group schemes, formal groups, and Tannakian duality.

Contribution

It characterizes affine functors as limits of affine schemes and establishes their duality properties, extending classical dualities to broader contexts.

Findings

Affine functors are equal to direct limits of affine schemes.

Affine schemes, formal schemes, and their completions are affine functors.

Endowing an affine functor with a monoid structure corresponds to a bialgebra structure.

Abstract

A functor of sets $\mathbb X$ over the category of $K$-commutative algebras is said to be an affine functor if its functor of functions, $\mathbb A_{\mathbb X}$, is reflexive and $\mathbb X=\Spec \mathbb A_{\mathbb X}$. We prove that affine functors are equal to a direct limit of affine schemes and that affine schemes, formal schemes, the completion of affine schemes along a closed subscheme, etc., are affine functors. Endowing an affine functor $\mathbb X$ with a functor of monoids structure is equivalent to endowing $\mathbb A_{\mathbb X}$ with a functor of bialgebras structure. If $\mathbb G$ is an affine functor of monoids, then $\mathbb A_{\mathbb G}^*$ is the enveloping functor of algebras of $\mathbb G$ and the category of $\mathbb G$-modules is equivalent to the category of $\mathbb A_{\mathbb G}^*$-modules. Applications of these results include Cartier duality, neutral Tannakian duality for affine group schemes, the equivalence between formal groups and Lie algebras in characteristic zero, etc.