Tate cohomology in axiomatic stable homotopy theory

Greenlees, J. P. C.

doi:10.1007/978-3-0348-8312-2_12

Cited by 20 publications

(17 citation statements)

References 29 publications

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“…This definition agrees with the construction of axiomatic Tate cohomology given in [Gre01], as there is a natural equivalence of cofiber sequences…”

Section: 2supporting

confidence: 76%

“…Our approach combines the abstract categorical framework of Hovey, Palmieri, and Strickland [HPS97] with the construction of torsion and completion functors for modules by Dwyer, Iyengar, and Greenlees [DG02, DGI06,Gre01]. Similar ideas have appeared in derived algebraic geometry [Lur11] and very recently in equivariant stable homotopy theory [MNN17].…”

Section: Introductionmentioning

confidence: 99%

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Local duality in algebra and topology

Barthel

Heard

Valenzuela

2018

Advances in Mathematics

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The first goal of this paper is to provide an abstract framework in which to formulate and study local duality in various algebraic and topological contexts. For any stable ∞-category C together with a collection of compact objects K ⊂ C we construct local cohomology and local homology functors satisfying an abstract version of local duality. When specialized to the derived category of a commutative ring A and a suitable ideal in A, we recover the classical local duality due to Grothendieck as well as generalizations by Greenlees and May. More generally, applying our result to the derived category of quasi-coherent sheaves on a quasi-compact and separated scheme X implies the local duality theorem of Alonso Tarrío, Jeremías López, and Lipman.As a second objective, we establish local duality for quasi-coherent sheaves over many algebraic stacks, in particular those arising naturally in stable homotopy theory. After constructing an appropriate model of the derived category in terms of comodules over a Hopf algebroid, we show that, in familiar cases, the resulting local cohomology and local homology theories coincide with functors previously studied by Hovey and Strickland. Furthermore, our framework applies to global and local stable homotopy theory, in a way which is compatible with the algebraic avatars of these theories. In order to aid computability, we provide spectral sequences relating the algebraic and topological local duality contexts.

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“…This definition agrees with the construction of axiomatic Tate cohomology given in [Gre01], as there is a natural equivalence of cofiber sequences…”

Section: 2supporting

confidence: 76%

Section: Introductionmentioning

confidence: 99%

Local duality in algebra and topology

Barthel

Heard

Valenzuela

2018

Advances in Mathematics

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“…In the second part of the paper, about the topology of Spc(SH(G) c ), we shall need additional tools relating to generalized Rickard idempotents in the sense of [BF11] and generalized Tate cohomology in the sense of Greenlees [Gre01].…”

Section: Tensor Idempotents and The Tate Constructionmentioning

confidence: 99%

The spectrum of the equivariant stable homotopy category of a finite group

Balmer¹,

Sanders

2016

Invent. math.

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We study the spectrum of prime ideals in the tensor-triangulated category of compact equivariant spectra over a finite group. We completely describe this spectrum as a set for all finite groups. We also make significant progress in determining its topology and obtain a complete answer for groups of square-free order. For general finite groups, we describe the topology up to an unresolved indeterminacy, which we reduce to the case of p-groups. We then translate the remaining unresolved question into a new chromatic blueshift phenomenon for Tate cohomology. Finally, we draw conclusions on the classification of thick tensor ideals.

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“…Some of the results in this paper, especially in §6, are related to results in [7], but here we take a different point of view. In the setting of commutative rings (or even schemes), the authors of [1] have already shown that A tors and A comp are equivalent categories; their approach does not involve the category of modules over End(A).…”

Section: Introductionmentioning

confidence: 97%

Complete modules and torsion modules

Dwyer¹,

Greenlees²

2002

ajm

126

173

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Abstract. Suppose that R is a ring and that A is a chain complex over R. Inside the derived category of differential graded R-modules there are naturally defined subcategories of A-torsion objects and of A-complete objects. Under a finiteness condition on A, we develop a Morita theory for these subcategories, find conceptual interpretations for some associated algebraic functors, and, in appropriate commutative situations, identify the associated functors as local homology or local cohomology. Some of the results are suprising even in the case R = Z and A = Z/p.

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Tate cohomology in axiomatic stable homotopy theory

Cited by 20 publications

References 29 publications

Local duality in algebra and topology

Local duality in algebra and topology

The spectrum of the equivariant stable homotopy category of a finite group

Complete modules and torsion modules

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