Locally coherent exact categories

Abstract

A locally coherent exact category is a finitely accessible additive category endowed with an exact structure in which the admissible short exact sequences are the directed colimits of admissible short exact sequences of finitely presentable objects. We show that any exact structure on a small idempotent-complete additive category extends uniquely to a locally coherent exact structure on the category of ind-objects; in particular, any finitely accessible category has the unique maximal and the unique minimal locally coherent exact category structures. All locally coherent exact categories are of Grothendieck type in the sense of Stovicek. We also discuss the canonical embedding of a small exact category into the abelian category of additive sheaves in connection with the locally coherent exact structure on the ind-objects, and deduce two periodicity theorems as applications.

LaTeX 2e with xy-pic; 34 pages, 21 commutative diagrams; v.2: small additions and corrections, references updated, Remark 1.3 inserted; v.3: Section 5 expanded, small things corrected, references updated; v.4: small things corrected, three paragraphs inserted in the beginning of the proof of Proposition 2.5

References in corpus (5)

• Singular compactness and definability for $\Sigma$-cotorsion and Gorenstein modules
• Covers, envelopes, and cotorsion theories in locally presentable abelian categories and contramodule categories
• Maximal exact structures on additive categories
• Periodic modules and acyclic complexes
• Fp-projective periodicity

Cited by in corpus (3)

• Flat comodules and contramodules as directed colimits, and cotorsion periodicity
• Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity
• Generalized periodicity theorems