Abelian envelopes of exact categories and highest weight categories

Kavli Affiliate: Alexey Bondal

| First 5 Authors: Agnieszka Bodzenta, Alexey Bondal, , ,

| Summary:

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.

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