Cyclotomic structure in the topological
Hochschild homology of DX

Malkiewich, Cary

doi:10.2140/agt.2017.17.2307

Cited by 11 publications

(25 citation statements)

References 25 publications

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“…Proof Part (i) is discussed in [, Proposition B.197; , § V.4], part (ii) is [, (B.198); , Proposition V.4.7], and part (iii) is [, (B.196)]. The uniqueness of the maps in (ii) and (iii) is from [, Theorem 3.20 and Remark 3.21] and the above observation that

Φ^{N}

may be interpreted in the ‘absolute’ sense.

□

…”

Section: Orthogonal G‐spectra and Geometric Fixed Pointsmentioning

confidence: 99%

“…We imagine these are arranged in a circle. Then the face maps multiply adjacent copies of

A

, the degeneracy maps insert copies of

A

along the unit

S \to A

, and the cyclic structure maps act by cyclic permutations (see, for example, [, § 2.1, , Section 4.2; , Definition 4.1.2).…”

Section: Topological Hochschild Homologymentioning

confidence: 99%

“…In fact, because

G

is abelian, it commutes with the action of

N

and so acts on

{normalΦ}^{N} X

. The action of

N ⩽ G

is trivial by [, Proposition 3.25], so

{normalΦ}_{abs}^{N} X

inherits a

G / N

‐action. As we explain in the proof of Lemma below,

{normalΦ}_{rel}^{N} X

and

{normalΦ}_{abs}^{N} X

are isomorphic as

G / N

‐spectra.…”

Section: Orthogonal G‐spectra and Geometric Fixed Pointsmentioning

confidence: 99%

“…We are therefore free to think of geometric fixed points in the ‘absolute’ or ‘relative’ sense without risk of confusion. This fact that the two notions of

Φ^{N}

coincide is also used implicitly in . Remark The property that

Φ^{N}

strictly commutes with forgetful functors holds more generally for certain subgroups of the

n

‐torus [, Lemmas 3.1.42 and 4.4.20].…”

Section: Orthogonal G‐spectra and Geometric Fixed Pointsmentioning

confidence: 99%

“…Definition (i)A pre‐cyclotomic spectrum is an orthogonal

S^{1}

‐spectrum

T

with choices of maps of

S^{1}

‐spectra

c_{r} : ρ_{r}^{*} {normalΦ}^{C_{r}} T \to T

for every integer

r ⩾ 1

such that following square commutes for any

r

and

s

Here

i t

is the map

c_{r, s}

from [, Proposition 2.6]; its definition is also forced from [, Theorem 3.23]. (ii)A cyclotomic spectrum is a pre‐cyclotomic spectrum

T

such that the induced map out of the derived geometric fixed points

ρ_{r}^{*} L {normalΦ}^{C_{r}} T \to ρ_{r}^{*} {normalΦ}^{C_{r}} T \overset{c_{r}}{⟶} T

is a stable equivalence of underlying orthogonal spectra for all

r ⩾ 1

. …”

Section: Cyclotomic Structuresmentioning

confidence: 99%

See 4 more Smart Citations

Comparing cyclotomic structures on different models for topological Hochschild homology

Dotto

Malkiewich

Patchkoria

et al. 2019

Journal of Topology

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The topological Hochschild homology THH(A) of an orthogonal ring spectrum A can be defined by evaluating the cyclic bar construction on A or by applying Bökstedt's original definition of prefixTHH to A. In this paper, we construct a chain of stable equivalences of cyclotomic spectra comparing these two models for THH(A). This implies that the two versions of topological cyclic homology resulting from these variants of THH(A) are equivalent.

show abstract

Φ^{N}

may be interpreted in the ‘absolute’ sense.

□

…”

Section: Orthogonal G‐spectra and Geometric Fixed Pointsmentioning

confidence: 99%

“…We imagine these are arranged in a circle. Then the face maps multiply adjacent copies of

A

, the degeneracy maps insert copies of

A

along the unit

S \to A

, and the cyclic structure maps act by cyclic permutations (see, for example, [, § 2.1, , Section 4.2; , Definition 4.1.2).…”

Section: Topological Hochschild Homologymentioning

confidence: 99%

“…In fact, because

G

is abelian, it commutes with the action of

N

and so acts on

{normalΦ}^{N} X

. The action of

N ⩽ G

is trivial by [, Proposition 3.25], so

{normalΦ}_{abs}^{N} X

inherits a

G / N

‐action. As we explain in the proof of Lemma below,

{normalΦ}_{rel}^{N} X

and

{normalΦ}_{abs}^{N} X

are isomorphic as

G / N

‐spectra.…”

Section: Orthogonal G‐spectra and Geometric Fixed Pointsmentioning

confidence: 99%

“…We are therefore free to think of geometric fixed points in the ‘absolute’ or ‘relative’ sense without risk of confusion. This fact that the two notions of