Homology, Homotopy and Applications

Volume 9 (2007)

Number 2

Heller triangulated categories

Pages: 233 – 320

DOI: https://dx.doi.org/10.4310/HHA.2007.v9.n2.a10


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


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.


triangulated category

2010 Mathematics Subject Classification


Published 1 January 2007