arXiv subjects

B. Shipley

Publications and source records attributed to B. Shipley.

7 recordsLinked to original sources

arXiv2018

An algebraic model for rational torus-equivariant spectra

We show that the category of rational G-spectra for a torus G is Quillen equivalent to an explicit small and practical algebraic model, thereby providing a universal de Rham model for rational G-equivariant cohomology theories. The result builds on the first author's Adams spectral sequence, the second author's functors making rational spectra algebraic. There are several steps, some perhaps of wider interest (1) isotropy separation (replacing the category of G-spectra by modules over a diagram of isotropically simple ring G-spectra) (2) passage to fixed points on ring and module categories (replacing diagrams of ring G-spectra by diagrams of ring spectra) (3) replacing diagrams of ring spectra by diagrams of differential graded algebras (4) rigidity (replacing diagrams of DGAs by diagrams of graded rings). Systematic use of cellularization of model categories is central.

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arXiv2015

Rational SO(2)-Equivariant Spectra

We prove that the category of rational SO(2)-equivariant spectra has a simple algebraic model. Furthermore, all of our model categories and Quillen equivalences are monoidal, so we can use this classification to understand ring spectra and module spectra via the algebraic model.

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arXiv2013

Fixed point adjunctions for equivariant module spectra

We consider the Quillen adjunction between fixed points and inflation in the context of equivariant module spectra over equivariant ring spectra, and give numerous examples including some based on geometric fixed points and some on the Eilenberg-Moore spectral sequence. These results were originally presented as part of our equivalence between rational torus-equivariant spectra and an algebraic model in arXiv:1101.2511. However, the present results apply in many other interesting cases explored here, which are not rational and where the ambient group is not a torus. The material in arXiv:1101.2511v3 will be revised to refer to this paper.

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arXiv2013

Homotopy theory of modules over diagrams of rings

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.

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arXiv2013

The cellularization principle for Quillen adjunctions

The Cellularization Principle states that under rather weak conditions a Quillen adjunction of stable model categories induces a Quillen equivalence on cellularizations provided there is a derived equivalence on cells. We give a proof together with a range of examples. The main result here was originally presented as an appendix of arXiv:1101.2511. However, the Cellularization Principle has many other applications which are explored here. The material in arXiv:1101.2511v3 will be revised to refer to this paper.

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arXiv2010

An algebraic model for free rational G-spectra for connected compact Lie groups G

We show that the category of free rational G-spectra for a connected compact Lie group G is Quillen equivalent to the category of torsion differential graded modules over the polynomial cohomology ring on the classifying space, H*(BG). The ingredients are enriched Morita equivalences, functors making rational spectra algebraic, and Koszul duality and thick subcategory arguments based on the simplicity of the derived category of a polynomial ring.

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arXiv2000

A telescope comparison lemma for THH

We extend to the non-connective case a lemma of Bokstedt about the equivalence of the telescope with a more complicated homotopy colimit of symmetric spectra used in the construction of THH.

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