On adjoint functors of the Heller operator

Abstract

Given an abelian category A with enough projectives, we can form its stable category _A_ := A/Proj(A)$. The Heller operator Omega : _A_ -> _A_ is characterised on an object X by a choice of a short exact sequence Omega X -> P -> X in A with P projective. If A is Frobenius, then Omega is an equivalence, hence has a left and a right adjoint. If A is hereditary, then Omega is zero, hence has a left and a right adjoint. In general, Omega is neither an equivalence nor zero. In the examples we have calculated via Magma, it has a left adjoint, but in general not a right adjoint. If A has projective covers, then Omega preserves monomorphisms; this would also follow from Omega having a left adjoint. I do not know an example where Omega does not have a left adjoint.