Homology, Homotopy and Applications

Volume 9 (2007)

Number 1

On lifting stable diagrams in Frobenius categories

Pages: 163 – 183

DOI: https://dx.doi.org/10.4310/HHA.2007.v9.n1.a6


Matthias Künzer (Lehrstuhl D für Mathematik, RWTH Aachen, Germany)


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.


Stable Frobenius category

2010 Mathematics Subject Classification


Published 1 January 2007