Homotopy theory of modules over diagrams of rings

J Greenlees,B Shipley

doi:10.1090/s2330-1511-2014-00012-2

Abstract

Given a diagram of rings, one may consider the category of modules over them. We are interested in the homotopy theory of categories of this type: given a suitable diagram of model categories M ( s ) \mathcal {M}(s) (as s s runs through the diagram), we consider the category of diagrams where the object X ( s ) X(s) at s s comes from M ( s ) \mathcal {M}(s) . We develop model structures on such categories of diagrams and Quillen adjunctions that relate categories based on different diagram shapes. Under certain conditions, cellularizations (or right Bousfield localizations) of these adjunctions induce Quillen equivalences. As an application we show that a cellularization of a category of modules over a diagram of ring spectra (or differential graded rings) is Quillen equivalent to modules over the associated inverse limit of the rings. Another application of the general machinery here is given in work by the authors on algebraic models of rational equivariant spectra. Some of this material originally appeared in the preprint “An algebraic model for rational torus-equivariant stable homotopy theory”, arXiv:1101.2511, but has been generalized here.

Highlights

Given a diagram of rings, one may consider the category of modules over them
We are interested in the homotopy theory of categories of this type: given a suitable diagram of model categories M(s) where functors relating them are left Quillen functors, we consider the category of diagrams where the object X(s) at s comes from M(s)
The purpose of this paper is to show that under suitable hypotheses, there are diagram-projective and diagram-injective model structures on the category (Theorem 3.1), and to investigate Quillen adjunctions associated to restricting the diagram (Theorems 5.3 and 5.5)

Summary

Introduction

Given a diagram of rings, one may consider the category of modules over them. We are interested in the homotopy theory of categories of this type: given a suitable diagram of model categories M(s) where functors relating them are left Quillen functors, we consider the category of diagrams where the object X(s) at s comes from M(s). Some of the necessary generality is slightly hidden here, since in the spectral part we must consider a context where the ring, and the group of equivariance varies with the position in the diagram With this generality, which motivates the setting of this current paper, we are able to describe the various models we use. Using the Cellularization Principle [10] (see Appendix A), we show that modules over the homotopy inverse limit of a given diagram of rings can be modelled by the cellularization of the category of modules over the diagram of rings (Proposition 4.1). None of the references [13], [16] or [3] consider changing diagram shapes, which is crucial in our application

Diagrams of rings and modules

Diagram-injective model structures

Inverse limit example

Adjunctions