[Paper Review] Homotopy theory of modules over diagrams of rings

J. P. C. Greenlees, Brooke Shipley|arXiv (Cornell University)|Sep 26, 2013

Algebraic structures and combinatorial models9 references3 citations

TL;DR

This paper establishes model structures for diagrams of modules over varying rings in a diagram category, using Quillen adjunctions and cellularization to show that the homotopy theory of modules over a diagram of ring spectra is Quillen equivalent to modules over the homotopy inverse limit of the diagram. The key contribution is a general framework for relating different diagram shapes via Quillen equivalences after cellularization, with applications to rational equivariant stable homotopy theory and the Hasse square construction.

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) (as s runs through the diagram), we consider the category of diagrams where the object X(s) at s comes from 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.

Motivation & Objective

• To develop model structures for categories of diagrams where each object comes from a different model category indexed by a diagram shape.
• To construct Quillen adjunctions between diagram categories of different shapes, particularly via restriction functors induced by subdiagrams.
• To show that under suitable conditions, cellularization of these adjunctions induces Quillen equivalences, thereby relating distinct homotopy theories.
• To apply the framework to show that modules over a diagram of ring spectra are Quillen equivalent to modules over the homotopy inverse limit of the diagram.
• To generalize techniques from rational torus-equivariant stable homotopy theory to a broader setting of diagrams of rings and modules.

Proposed method

• Construct diagram-projective and diagram-injective model structures on categories of diagrams indexed by a small category, where each vertex category is a model category.
• Define restriction functors between diagram categories induced by inclusions of diagram shapes, leading to Quillen adjunctions.
• Apply the Cellularization Principle to show that after cellularization, Quillen adjunctions become Quillen equivalences when derived functors preserve small objects and weak equivalences on cells.
• Use derived functors and homotopy limits to relate modules over a diagram of rings to modules over the inverse limit of the diagram.
• Leverage stable model category theory and smallness conditions to ensure cellularizations preserve Quillen equivalences.
• Apply results to the Hasse square example, showing that modules over the diagram of rings are Quillen equivalent to modules over the inverse limit (e.g., ℤ) after cellularization.

Experimental results

Research questions

• RQ1 Under what conditions does a Quillen adjunction between diagram categories of modules over diagrams of rings become a Quillen equivalence after cellularization?
• RQ2 How can one relate the homotopy theory of modules over a diagram of ring spectra to the homotopy theory of modules over the homotopy inverse limit of the diagram?
• RQ3 What conditions ensure that the derived functors of restriction and extension of scalars preserve small objects and weak equivalences in the cellularization process?
• RQ4 Can the Cellularization Principle be applied to diagrams of model categories where the categories vary with the diagram position?
• RQ5 In what way does changing the diagram shape (via subdiagram inclusions) affect the resulting homotopy theory of modules, and when are the resulting categories Quillen equivalent?

Key findings

• The paper constructs diagram-projective and diagram-injective model structures on categories of diagrams of objects from varying model categories indexed by a small category.
• It establishes Quillen adjunctions between diagram categories induced by inclusions of diagram shapes, particularly via restriction functors.
• After cellularization, these Quillen adjunctions become Quillen equivalences when the derived functors preserve small objects and weak equivalences on the chosen cells.
• The main application shows that the category of modules over a diagram of ring spectra is Quillen equivalent to the category of modules over the homotopy inverse limit of the diagram, via cellularization.
• The Hasse square example is shown to satisfy the conditions, so modules over the diagram of ℚ, ℤ_p, and their tensor products are Quillen equivalent to modules over ℤ after cellularization.
• The framework generalizes earlier results in rational torus-equivariant stable homotopy theory and provides a systematic method for constructing algebraic models of stable homotopy categories.