The Cellularization Principle for Quillen adjunctions

Greenlees, J. P. C.; Shipley, Brooke

doi:10.4310/hha.2013.v15.n2.a11

Cited by 19 publications

(44 citation statements)

References 16 publications

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“…This is a model for the more general adjunctions considered in Section 5. The particular case of a pullback diagram of rings, such as in the classical Hasse principle, is treated in more detail in Section 6 of [10]. Here we will work in the context of ring and module spectra, but this material also easily translates to the differential graded context.…”

Section: Inverse Limit Examplementioning

confidence: 99%

“…Since finite limits are constructed from finite products and equalizers, this shows that finite limits commute with infinite coproducts. This argument is worked out in more detail for D a pullback diagram in Section 6.4 of [10].…”

Section: R-cell-r-mod Q Fr-cell-fr-modmentioning

confidence: 99%

“…We show that the adjunction associated to the change from the diagram of rings to the one homotopy inverse limit ring induces a Quillen equivalence after applying the Cellularization Principle from [10]; see also Proposition A.1. This is a model for the more general adjunctions considered in Section 5.…”

Section: Inverse Limit Examplementioning

confidence: 99%

“…A particularly well-known example is (differential graded) modules over the classical Hasse square which considers the integers as the pull-back ring of the rationals, the p-adic integers for all primes p, and their tensor products. Using the Cellularization Principle [10] (see also 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). This is the model category version of the local to global principle for the Hasse square.…”

Section: Amentioning

confidence: 99%

“…To relate categories based on these two diagram shapes we construct a larger diagram shape category which contains the smaller ones, so that a diagram based on the new larger shape restricts to two smaller diagrams of the original shapes. We then show that a suitable inclusion of diagram shapes induces a Quillen adjunction, and apply the Cellularization Principle [10] to show that it induces a Quillen equivalence after cellularization.…”

Section: Amentioning

confidence: 99%

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Homotopy theory of modules over diagrams of rings

Greenlees¹,

Shipley²

2014

Proc. Amer. Math. Soc. Ser. B

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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.

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Section: Inverse Limit Examplementioning

confidence: 99%

Section: R-cell-r-mod Q Fr-cell-fr-modmentioning

confidence: 99%

Section: Inverse Limit Examplementioning

confidence: 99%

Section: Amentioning

confidence: 99%

Section: Amentioning

confidence: 99%

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Homotopy theory of modules over diagrams of rings

Greenlees¹,

Shipley²

2014

Proc. Amer. Math. Soc. Ser. B

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An algebraic model for rational naïve-commutative ring SO(2)-spectra and equivariant elliptic cohomology

2020

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Equipping a non-equivariant topological $$\text {E}_\infty $$ E ∞ -operad with the trivial G-action gives an operad in G-spaces. For a G-spectrum, being an algebra over this operad does not provide any multiplicative norm maps on homotopy groups. Algebras over this operad are called naïve-commutative ring G-spectra. In this paper we take $$G=SO(2)$$ G = S O ( 2 ) and we show that commutative algebras in the algebraic model for rational SO(2)-spectra model rational naïve-commutative ring SO(2)-spectra. In particular, this applies to show that the SO(2)-equivariant cohomology associated to an elliptic curve C of Greenlees (Topology 44(6):1213–1279, 2005) is represented by an $$\text {E}_\infty $$ E ∞ -ring spectrum. Moreover, the category of modules over that $$\text {E}_\infty $$ E ∞ -ring spectrum is equivalent to the derived category of sheaves over the elliptic curve C with the Zariski torsion point topology.

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On Solid and Rigid Monoids in Monoidal Categories

Gutiérrez

2014

Appl Categor Struct

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We introduce the notion of solid monoid and rigid monoid in monoidal categories and study the formal properties of these objects in this framework. We show that there is a one to one correspondence between solid monoids, smashing localizations and mapping colocalizations, and prove that rigid monoids appear as localizations of the unit of the monoidal structure. As an application, we study solid and rigid ring spectra in the stable homotopy category and characterize connective solid ring spectra as Moore spectra of subrings of the rationals. p∈P Z p , where P is any set of primes. However, there are groups such as the Prüfer group Z/p ∞ or the p-adic field Q p that do not admit a rigid ring structure.

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The Cellularization Principle for Quillen adjunctions

Abstract: 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.

Cited by 19 publications

References 16 publications

Homotopy theory of modules over diagrams of rings

Homotopy theory of modules over diagrams of rings

An algebraic model for rational naïve-commutative ring SO(2)-spectra and equivariant elliptic cohomology

On Solid and Rigid Monoids in Monoidal Categories

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