Spectral Mackey functors and equivariant algebraic K-theory (I)

Barwick, Clark

doi:10.1016/j.aim.2016.08.043

Cited by 101 publications

(133 citation statements)

References 18 publications

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“…Remark 2.17. One equivalent description of TSp gen p following Barwick [Bar17] is as the ∞-category of product-preserving functors from the effective Burnside category of the orbit category {S 1 /C p n } n 0 ⊆ S BS 1 to spectra. In this language, (TSp gen p ) 0 is equivalent to the ∞-category of product-preserving functors from the Burnside category to Sp 0 and similarly for the ∞-category of coconnective objects.…”

Section: The Genuine Cyclotomic T-structurementioning

confidence: 99%

Cartier modules and cyclotomic spectra

Antieau

Nikolaus

2020

J. Amer. Math. Soc.

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We construct and study a t-structure on p-typical cyclotomic spectra and explain how to recover crystalline cohomology of smooth schemes over perfect fields using this t-structure. Our main tool is a new approach to p-typical cyclotomic spectra via objects we call p-typical topological Cartier modules. Using these, we prove that the heart of the cyclotomic t-structure is the full subcategory of derived V -complete objects in the abelian category of p-typical Cartier modules.We say that a p-typicalThe existence and uniqueness of such a t-structure is a formal consequence of the fact that (CycSp p ) 0 is presentable and is closed under colimits and extensions in CycSp p . The difficult part of the theorem is the identification of the heart.Recall Bökstedt's theorem, which says that π * THH(F p ) = F p [b], a polynomial ring on a degree 2 generator. More generally, using the vanishing of the cotangent complex, one deduces that π * THH(k) = k[b] for any perfect ring k. Our interest in the cyclotomic t-structure was piqued by the discovery of the next result. 1 Unless otherwise specified, all quotients M/V n are computed in the derived sense (and hence are given as the cofiber of M V n −− → M in the derived category) as are all limits.Corollary 10. If X is a bounded below p-typical cyclotomic spectrum, then the natural S 1 -equivariant map TR(X) → X induces a p-adic equivalence TR(X) tS 1 → X tS 1 .Proof. By Theorem 9, the counit map TR(X)/TR(X) hCp → X is an equivalence. In particular, we have a cofiber sequence TR(X) hCp → TR(X) → X.Applying (−) tS 1 , we obtain a cofiber sequence p-adically equivalent to[NS17, Lemma II.4.2]. But, (TR(X) hCp ) tCp ≃ 0 by the Tate orbit lemma [NS17, Lemma I.2.1].2 In early talks on this project, we called these topological Dieudonné modules. The cyclotomic t-structureIn this section we define the cyclotomic t-structure for integral and p-typical cyclotomic spectra in their genuine and non-genuine flavors. With some difficulty, one can prove some basic facts about truncations in

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Section: The Genuine Cyclotomic T-structurementioning

confidence: 99%

Cartier modules and cyclotomic spectra

Antieau

Nikolaus

2020

J. Amer. Math. Soc.

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“…We can define Mackey functors on finGpd in the same way. This kind of generalization of a Mackey functor onto higher categories can be also found in [2]. By Remark 1.21, the category SMack (finGpd) = SMack (C ′ ) becomes equivalent to SMack (C ), and Mack (…”

Section: Introductionmentioning

confidence: 97%

“…However, we remark that (iii) is stronger than (Der4g), since we do not assume (Der2). 2 As the proof suggests, the existence of an isomorphism (p J ) ! • p * I ∼ = J * • I !…”

Section: [Compatibility With the Contravariant Parts]mentioning

confidence: 99%

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Biset Functors as Module Mackey Functors and its Relation to Derivators

Nakaoka

2016

Communications in Algebra

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Abstract. In this article, we will show that the category of biset functors can be regarded as a reflective monoidal subcategory of the category of Mackey functors on the 2-category of finite groupoids. This reflective subcategory is equivalent to the category of modules over the Burnside functor. As a consequence of the reflectivity, we can associate a biset functor to any derivator on the 2-category of finite

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“…To explain what we mean by cyclotomic here, we must first describe geometric fixed points in the derived category of Mackey functors. This is related to Remark 6.5 of [19] and B.7 of [3]. We will define our version in terms of a particular choice of point-set model for the derived functor.…”

Section: Geometric Fixed Points In Mackey Functorsmentioning

confidence: 99%

The Witt vectors for Green functors

et al. 2019

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We define twisted Hochschild homology for Green functors. This construction is the algebraic analogue of the relative topological Hochschild homology THH Cn (−), and under flatness conditions it describes the E 2 term of the Künneth spectral sequence for relative THH. Applied to ordinary rings, we obtain new algebraic invariants. Extending Hesselholt's construction of the Witt vectors of noncommutative rings, we interpret our construction as providing Witt vectors for Green functors.

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Spectral Mackey functors and equivariant algebraic K-theory (I)

Cited by 101 publications

References 18 publications

Cartier modules and cyclotomic spectra

Cartier modules and cyclotomic spectra

Biset Functors as Module Mackey Functors and its Relation to Derivators

The Witt vectors for Green functors

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