On lifting stable diagrams in Frobenius categories

Abstract

Suppose given a Frobenius category E, i.e. an exact category with a big enough subcategoryB of bijectives. LetE :=E=B denote its classical stable category. For example, we may take E to be the category of complexes C(A) with entries in an additive category A, in which case E is the homotopy category of complexes K(A). Suppose given a nite poset D that satises the combinatorial condition of being ind-at . Then, given a diagram of shape D with values inE (i.e. stably commutative), there exists a diagram consisting of pure monomorphisms with values inE (i.e. commutative) that is isomorphic, as a diagram with values inE, to the given diagram.