Complete cotorsion pairs in exact categories
Zhi-Wei Li
TL;DR
This paper establishes conditions under which cotorsion pairs are complete in exact categories, extending existing criteria and providing examples in chain complex categories.
Contribution
It generalizes the completeness criterion for cotorsion pairs in exact categories using the generalized small object argument, unifying and extending prior results.
Findings
Cotorsion pairs cogenerated by a class are complete under certain conditions.
The generalized small object argument can be applied to arbitrary exact categories.
Examples are provided in categories of chain complexes of exact categories.
Abstract
We show a cotorsion pair cogenerated by a class is complete under suitable conditions in an arbitrary exact category using the generalized small object argument given by Chorny. This recovers Saor\'in and \v{S}\v{t}ov\'{i}\v{c}ek's criterion of the completeness of cotorsion pairs in their efficient exact categories. Examples in the categories of chain complexes of exact categories are given.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Nonlinear Waves and Solitons