A poisonous example to explicit resolutions of unbounded complexes

TL;DR

This paper demonstrates the failure of common methods for constructing resolutions of unbounded complexes, proposes a condition to fix this, and explores model structures in relative homological algebra.

Contribution

It identifies limitations of explicit resolution methods and introduces the (Ab.4$^*$)-$k$ axiom as a solution, extending to relative homological algebra frameworks.

Findings

Methods fail on simple explicit complexes

(Ab.4$^*$)-$k$ axiom rescues resolution techniques

Model structures exist under the (Ab.4$^*$)-$k$ condition

Abstract

We show that various methods for explicitly building resolutions of unbounded complexes in fact fail when applied to a rather simple and explicit complex. We show that one way to rescue these methods is to assume Roos (Ab.4$^*$)-$k$ axiom, which we adapt to encompass also resolutions in the framework of relative homological algebra. In the end we discuss the existence of model structures for relative homological algebra for unbounded complex under the relative (Ab.4$^*$)-$k$ condition, and present a variety of examples where our results apply.