Model category structures and spectral sequences

TL;DR

This paper constructs and compares model category structures on filtered complexes and bicomplexes over a commutative ring, based on spectral sequence stages, and explores their relationships via shift and décalage functors.

Contribution

It introduces cofibrantly generated model structures on these categories where weak equivalences are spectral sequence stage quasi-isomorphisms, and analyzes their interrelations.

Findings

Model structures are established for filtered complexes and bicomplexes.

Different model structures are related through shift and décalage functors.

The framework facilitates homotopical analysis of spectral sequence stages.

Abstract

Let k be a commutative ring with unit. We endow the categories of filtered complexes and of bicomplexes of k-modules, with cofibrantly generated model structures, where the class of weak equivalences is given by those morphisms inducing a quasi-isomorphism at a certain fixed stage of the associated spectral sequence. For filtered complexes, we relate the different model structures obtained, when we vary the stage of the spectral sequence, using the functors shift and d\'ecalage.