On topological Hochschild homology of the K(1)‐local sphere

Angelini-Knoll, Gabe

doi:10.1112/topo.12182

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Thr Of Projective Spaces1

Definition Of Thr1

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2021

2023

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Cited by 3 publications

(2 citation statements)

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“…Together with equation (5.6), we obtain an equivalence THR(P n S ) 1) if n is odd. Use Propositions 4.2.15 and 5.2.5 for the standard cover of P n to obtain an equivalence THR(X × P n ) THR(X) ∧ THR(P n S ) whenever X is an affine scheme with involution.…”

Section: Thr Of Projective Spacesmentioning

confidence: 90%

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Real Topological Hochschild Homology of Schemes

Hornbostel

Park²

2023

J. Inst. Math. Jussieu

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Section: Thr Of Projective Spacesmentioning

confidence: 90%

“…65]. Note also that it is possible to define THR for Z/2-spectra with slightly less structure, for example, for E σ -algebras as in [1].…”

Section: Definition Of Thrmentioning

confidence: 99%

Real Topological Hochschild Homology of Schemes

Hornbostel

Park²

2023

J. Inst. Math. Jussieu

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Detecting 𝛽 elements in iterated algebraic K-theory

Angelini-Knoll

2023

Trans. Amer. Math. Soc.

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The Lichtenbaum–Quillen conjecture (LQC) relates special values of zeta functions to algebraic K-theory groups. The Ausoni–Rognes red-shift conjectures generalize the LQC to higher chromatic heights in a precise sense. In this paper, we propose an alternate generalization of the LQC to higher chromatic heights and give evidence for it at height two. In particular, if the n n -th Greek letter family is detected by a commutative ring spectrum R R , then we conjecture that the n + 1 n+1 -st Greek letter family will be detected by the algebraic K-theory of R R . We prove this in the case n = 1 n=1 for R = K ( F q ) R=\mathrm {K}(\mathbb {F}_q) modulo ( p , v 1 ) (p,v_1) where p ≥ 5 p\ge 5 and q = ℓ k q=\ell ^k is a prime power generator of the units in Z / p 2 Z \mathbb {Z}/p^2\mathbb {Z} . In particular, we prove that the commutative ring spectrum K ( K ( F q ) ) \mathrm {K}(\mathrm {K}(\mathbb {F}_q)) detects the part of the p p -primary β \beta -family that survives mod ( p , v 1 ) (p,v_1) . The method of proof also implies that these β \beta elements are detected in iterated algebraic K-theory of the integers. Consequently, one may relate iterated algebraic K-theory groups of the integers to integral modular forms satisfying certain congruences.

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The topological Hochschild homology of algebraic K-theory of finite fields

Höning

2021

Ann. K-Th.

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On topological Hochschild homology of the K(1)‐local sphere

Abstract: We compute mod (p,v1) topological Hochschild homology of the connective cover of the K(1)‐local sphere spectrum for all primes p⩾3. This is accomplished using a May‐type spectral sequence in topological Hochschild homology constructed from a filtration of a commutative ring spectrum.

Cited by 3 publications

References 29 publications

Real Topological Hochschild Homology of Schemes

Real Topological Hochschild Homology of Schemes

Detecting 𝛽 elements in iterated algebraic K-theory

The topological Hochschild homology of algebraic K-theory of finite fields

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