Comparing cyclotomic structures on different models for topological Hochschild homology

Dotto, Emanuele; Malkiewich, Cary; Patchkoria, Irakli; Sagave, Steffen; Woo, Calvin

doi:10.1112/topo.12116

Cited by 10 publications

(11 citation statements)

References 27 publications

(122 reference statements)

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“…In this section, we compare the constructions of §III.2 and §III.5. A similar comparison between the Bökstedt model for THH and the cyclic bar construction model for THH (in the model category of [6]) will also appear in [29].…”

Section: Iii6 Comparisonmentioning

confidence: 97%

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On topological cyclic homology

2018

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204t. nikolaus and p. scholze t. nikolaus and p. scholze the analogy between TC and syntomic cohomology has been known, and pursued for example by Kaledin,[57],[58]; see also at the end of the introduction of [13] for an early suggestion of such a relation. As explained to us by Kaledin, our main theorem is closely related to his results relating cyclotomic complexes and filtered Dieudonné modules.By Theorem 1.4, the information stored in the genuine cyclotomic spectrum (THH(A), (Φ p ) p ) can be characterized explicitly. In order for this to be useful, however, we need to give a direct construction of this information. In other words, for A∈Alg E1 (Sp), we have to define directly a T/C p ∼ =T-equivariant Frobenius mapWe will give two discussions of this, first for associative algebras, and then indicate a much more direct construction for E ∞ -algebras.Let us discuss the associative case for simplicity for p=2. Note that, by definition, the source THH(A) is the realization of the cyclic spectrum

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Section: Iii6 Comparisonmentioning

confidence: 97%

“…Definition III.4.3 below. ( 4 ) The relation between these constructions is the subject of current investigations, and we refer to [6] and [29] for a discussion of this point. In the following, we denote by THH(Ã) the realization of the cyclic object defined through the Bökstedt construction.…”

Section: Introductionmentioning

confidence: 99%

On topological cyclic homology

2018

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“…Then [38,Corollary 7.3] implies that the definition of TCR(A; p) as the homotopy equalizer of the diagram (9) and the definition of Real topological cyclic homology TCR(A; p) of Definition 3.27 agree after p-completion.…”

Section: 3mentioning

confidence: 99%

Real topological Hochschild homology via the norm and Real Witt vectors

Angelini-Knoll¹,

Gerhardt²,

Hill³

2021

Preprint

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We prove that Real topological Hochschild homology can be characterized as the norm from the cyclic group of order 2 to the orthogonal group O(2). From this perspective, we then prove a multiplicative double coset formula for the restriction of this norm to dihedral groups of order 2m. This informs our new definition of Real Hochschild homology of rings with anti-involution, which we show is the algebraic analogue of Real topological Hochschild homology. Using extra structure on Real Hochschild homology, we define a new theory of p-typical Witt vectors of rings with anti-involution. We end with an explicit computation of the degree zero D2m-Mackey functor homotopy groups of THR(Z) for m odd. This uses a Tambara reciprocity formula for sums for general finite groups, which may be of independent interest.

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“…[19] and Hesselholt-Madsen [41] prove that THH(R) admits the structure of a genuine cyclotomic spectrum by using the Bökstedt construction. Angeltveit-Blumberg-Gerhardt-Hill-Lawson-Mandell [1] construct the genuine cyclotomic structure on THH(R) using the Hill-Hopkins-Ravenel norm [45], and these constructions are equivalent by the work of Dotto-Malkiewich-Patchkoria-Sagave-Woo [27]. Nikolaus-Scholze [59] construct a cyclotomic structure on THH(R) using the Tatevalued diagonal.…”

Section: Remark 333 There Is a Canonical Sequence Of Functors Of ∞-Ca...mentioning

confidence: 99%

On curves in K-theory and TR

McCandless¹

2021

Preprint

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We prove that TR is corepresentable by the reduced topological Hochschild homology of the flat affine line S[t] as a functor defined on the ∞-category of cyclotomic spectra taking values in the ∞-category of cyclotomic spectra with Frobenius lifts, refining a result of Blumberg-Mandell. To that end, we define the notion of an integral topological Cartier module using Barwick's formalism of Mackey functors on orbital ∞-categories, extending the work of Antieau-Nikolaus in the p-typical case. As an application, we show that TR evaluated on a connective E 1 -ring admits a description in terms of the spectrum of curves on algebraic K-theory generalizing the work of Hesselholt and Betley-Schlichtkrull. Contents 1. Introduction 1 1.1. Statement of results 2 1.2. Methods 3 2. Cyclotomic spectra with Frobenius lifts and TR 6 2.1. The epicyclic category 6 2.2. Spaces with Frobenius lifts 11 2.3. Cyclotomic spectra with Frobenius lifts 15 2.4. Topological restriction homology 18 3. Comparison with genuine TR 20 3.1. Equivariant stable homotopy theory 20 3.2. Topological Cartier modules 24 3.3. Genuine cyclotomic spectra and genuine TR 30 4. Applications to curves on K-theory 34 4.1. Topological Hochschild homology of truncated polynomial algebras 34 4.2. Curves on K-theory 42 References 45

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Comparing cyclotomic structures on different models for topological Hochschild homology

Cited by 10 publications

References 27 publications

On topological cyclic homology

On topological cyclic homology

Real topological Hochschild homology via the norm and Real Witt vectors

On curves in K-theory and TR

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