Abelian envelopes of exact categories and highest weight categories

TL;DR

This paper introduces the concepts of abelian envelopes and thin exact categories, establishing their properties and connections to highest weight categories, and interprets Ringel duality through these frameworks.

Contribution

It develops the theory of abelian envelopes and thin exact categories, linking them to highest weight categories and providing a new perspective on Ringel duality.

Findings

Existence of right and left abelian envelopes for certain exact categories.

Highest weight categories are characterized as abelian envelopes of thin categories.

Ringel duality is interpreted as a duality between abelian envelopes via derived categories.

Abstract

We define admissible and weakly admissible subcategories in exact categories and prove that the former induce semi-orthogonal decompositions on the derived categories. We develop the theory of thin exact categories, an exact-category analogue of triangulated categories generated by exceptional collections. The right and left abelian envelopes of exact categories are introduced, an example being the category of coherent sheaves on a scheme as the right envelope of the category of vector bundles. The existence of right (left) abelian envelopes is proved for exact categories with projectively (injectively) generating subcategories with weak (co)kernels. We show that highest weight categories are precisely the right/left envelopes of thin categories. Ringel duality is interpreted as a duality between the right and left abelian envelopes of a thin exact category. The duality for thin exact categories is introduced by means of derived categories and Serre functor on them.