A poisonous example to explicit resolutions of unbounded complexes
Dolors Herbera, Wolfgang Pitsch, Manuel Saorín, Simone Virili
TL;DR
This work reveals that explicit dg-injective resolutions for unbounded complexes can fail in simple settings, exposing the fragility of several classical constructions. It identifies the Roos $(Ab.4^*)$-$k$ axiom, and its relative versions, as the essential hypothesis that salvages these methods and enables robust relative homological machinery. Through concrete counterexamples (notably a Nagata-based Grothendieck category and a specially crafted unbounded complex) and systematic analysis of Spaltenstein towers, Cartan–Eilenberg resolutions, and Saneblidze multicomplexes, the paper delineates which approaches succeed under $(Ab.4^*)$-$k$ and which do not. By developing an $\mathcal{I}$-injective model structure on chain complexes relative to an injective class $\mathcal{I}$, and connecting these to Verdier quotients and derived categories, the authors provide a flexible framework for relative derived categories and applications to tilting theory and quasi-coherent sheaves. The results have concrete implications for constructing explicit resolutions, understanding when such constructions are valid, and applying the theory to geometric contexts and tilting-cotorsion frameworks.
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.