On adjoint functors of the Heller operator

TL;DR

This paper investigates the adjoint functors of the Heller operator in stable categories derived from abelian categories, revealing conditions under which these functors exist and their properties.

Contribution

It characterizes when the Heller operator has adjoints in stable categories, especially in Frobenius and hereditary cases, and explores the existence of left adjoints in general.

Findings

Omega is an equivalence in Frobenius categories.

Omega is zero in hereditary categories.

In general, Omega often has a left adjoint but not a right adjoint.

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.