Real topological Hochschild homology

Dotto, Emanuele; Moi, Kristian; Patchkoria, Irakli; Reeh, Sune Precht

doi:10.48550/arxiv.1711.10226

Cited by 5 publications

(9 citation statements)

References 17 publications

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“…. , k} (see [DMPPR17] and [Høg16]). For every m ≥ 0 one can define a Z 2-spectrum TRR m+1 (E; p) ∶= THR(E) C p m , which is a Z 2-equivariant refinement of the TR-spectra of [BHM93].…”

Section: Introductionmentioning

confidence: 99%

“…For every m ≥ 0 one can define a Z 2-spectrum TRR m+1 (E; p) ∶= THR(E) C p m , which is a Z 2-equivariant refinement of the TR-spectra of [BHM93]. In Corollary 5.2 of [DMPPR17] we identify the Z 2-Tambara functor of components π 0 THR(E), in the case m = 0. Theorem [HM97] of Hesselholt and Madsen asserts that the components of the underlying ring spectrum π 0 TR m+1 (E; p) = π 0 THH(E) C p m ≅ W m+1 (π 0 E; p) = W m+1 (π 0 THH(E); p) are naturally isomorphic to the ring of p-typical (m + 1)-truncated Witt vectors of π 0 E. Here we establish a real version of this statement for odd primes.…”

Section: Introductionmentioning

confidence: 99%

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Witt Vectors, Polynomial Maps, and Real Topological Hochschild Homology

Dotto¹,

Moi²,

Patchkoria³

2019

Preprint

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We show that various flavors of Witt vectors are functorial with respect to multiplicative polynomial laws of finite degree. We then deduce that the p-typical Witt vectors are functorial in multiplicative polynomial maps of degree at most p − 1. This extra functoriality allows us to extend the p-typical Witt vectors functor from commutative rings to Z 2-Tambara functors, for odd primes p. We use these Witt vectors for Tambara functors to describe the components of the dihedral fixed-points of the real topological Hochschild homology spectrum at odd primes.

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“…. , k} (see [DMPPR17] and [Høg16]). For every m ≥ 0 one can define a Z 2-spectrum TRR m+1 (E; p) ∶= THR(E) C p m , which is a Z 2-equivariant refinement of the TR-spectra of [BHM93].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Witt Vectors, Polynomial Maps, and Real Topological Hochschild Homology

Dotto¹,

Moi²,

Patchkoria³

2019

Preprint

Self Cite

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“…Remark 3.4.1. It is possible to identify N λ HF 2 for the dihedral groups D 2 n with the genuine action of D 2 n on THR(HF p ) (see [Dot+17;HM15;QS19]). An approach similar to that in Proposition 3.4 should prove the analogous result for the dihedral groups.…”

Section: Mackey Functorsmentioning

confidence: 99%

Eilenberg Mac Lane spectra as p-cyclonic Thom spectra

Levy¹

2021

Preprint

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Hopkins and Mahowald gave a simple description of the mod p Eilenberg Mac Lane spectrum HF p as the free E 2 -algebra with an equivalence of p and 0. We show for each faithful 2-dimensional representation λ of a p-group G that the G-equivariant Eilenberg Mac Lane spectrum HF p is the free E λ -algebra with an equivalence of p and 0. This unifies and simplifies recent work of Behrens, Hahn, and Wilson, and extends it to include the dihedral 2-subgroups of O(2). The main new idea is that HF p has a simple description as a p-cyclonic module over THH(HF p ). We show our result is the best possible one in that it gives all groups G and representations V such that HF p is the free E V -algebra with an equivalence of p and 0.

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“…Moreover, the same idea of adding a "twisted" C 2 -action also works for the cyclic bar construction, which provides one definition for THR, the real topological Hochschild homology (see [DMPR17] section 2). Section 4 in [Dotto12] also explains how the C 2 -equivariant delooping machine acts on THR.…”

Section: Introductionmentioning

confidence: 99%