On lifting stable diagrams in Frobenius categories

Suppose given a Frobenius category ${\cal E}$, i.e. an exact category with a big enough subcategory ${\cal B}$ of bijectives. Let $\underline{\cal E} := {\cal E}/{\cal B}$ denote its classical stable category. For example, we may take ${\cal E}$ to be the category of complexes $\mathrm{C}({\cal A})$ with entries in an additive category ${\cal A}$, in which case $\underline{\cal E}$ is the homotopy category of complexes $\mathrm{K}({\cal A})$. Suppose given a finite poset $D$ that satisfies the combinatorial condition of being ind-flat. Then, given a diagram of shape $D$ with values in $\underline{\cal E}$ (i.e. stably commutative), there exists a diagram consisting of pure monomorphisms with values in ${\cal E}$ (i.e. commutative) that is isomorphic, as a diagram with values in $\underline{\cal E}$, to the given diagram.

Keywords

Stable Frobenius category

2010 Mathematics Subject Classification

18E10

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Published 1 January 2007