Heller triangulated categories

Let ${\cal E}$ be a Frobenius category. Let $\underline{\cal E}$ denote its stable category. The shift functor on $\underline{\cal E}$ induces, by pointwise application, an inner shift functor on the category of acyclic complexes with entries in $\underline{\cal E}$. Shifting a complex by $3$ positions yields an outer shift functor on this category. Passing to quotient modulo split acyclic complexes, Heller remarked that inner and outer shift become isomorphic, via an isomorphism satisfying yet a further compatibility. Moreover, Heller remarked that a choice of such an isomorphism determines a Verdier triangulation on $\underline{\cal E}$, except for the octahedral axiom. We generalise the notion of acyclic complexes such that the accordingly enlarged version of Heller’s construction includes octahedra.

Keywords

triangulated category

2010 Mathematics Subject Classification

18E30

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Published 1 January 2007