Anomaly in exact categories

Abelian envelopes of exact categories and highest weight 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 rightenvelope 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 define admissible and weakly admissible subcategories in exact categories and prove that the former induce semi-orthogonal decompositions on the derived categories.We show that highest weight categories are precisely the right/left envelopes of thin categories.Thin exact categories, right and left abelian envelopes, semi-orthogonal decompositions.