Detecting periodic elements in higher topological Hochschild homology

Veen, Torleif

doi:10.2140/gt.2018.22.693

Cited by 14 publications

(15 citation statements)

References 25 publications

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“…Proof. We learned of results similar to the following from [Vee18]: if S → A → B are cofibrations of commutative S-algebras, then there is a weak equivalence…”

Section: σY ⊗ Kumentioning

confidence: 94%

“…After having proved the theorem, though, we have that KU does satisfy this for the special case of (6.15). The partial results of [Vee18] (extended by [BLP + 15]) prove that A = HF p also satisfy it for (6.15), at least in a certain range relating n and p. We are led to ask ourselves the question, as [DT18, 4.1] did for A = HF p , of whether more generally KU is such that X ⊗ KU Y ⊗ KU provided ΣX ΣY . More ambitiously, it would be interesting to find conditions on any commutative S-algebra A that guarantee this property.…”

Section: Iterated Topological Hochschild Homology Of Kumentioning

confidence: 99%

“…Spheres S n give a topological version of Pirashvili's higher order Hochschild homology of commutative rings [Pir00]. Complete calculations of these invariants for a given R are scarce: see for example [Sch11] for the case of spectra, [Vee18] and [BLP + 15] for partial computations for the Eilenberg-Mac Lane ring spectrum HF p of the field with p elements and other related ring spectra, and [DLR18] for Eilenberg-Mac Lane spectra of some rings of integers.…”

Section: Introductionmentioning

confidence: 99%

“…of whether − ⊗ R is a stable invariant. This turns out not to be true in general [DT18], but the computations of [Vee18] show that, in a certain range relating n and p, X ⊗ HF p Y ⊗ HF p when X = T n and Y = n i=1 (S i ) ∨( n i ) . Our results show that the same is true for R = KU , see…”

Section: Introductionmentioning

confidence: 99%

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Higher topological Hochschild homology of periodic complex K-theory

Stonek

2020

Topology and its Applications

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We describe the topological Hochschild homology of the periodic complex K-theory spectrum, T HH(KU), as a commutative KU-algebra: it is equivalent to KU [K(Z, 3)] and to F (ΣKU Q), where F is the free commutative KU-algebra functor on a KU-module. Moreover, F (ΣKU Q) KU ∨ ΣKU Q , a square-zero extension. In order to prove these results, we first establish that topological Hochschild homology commutes, as an algebra, with localization at an element. Then, we prove that T HH n (KU), the n-fold iteration of T HH(KU), i.e. T n ⊗ KU , is equivalent to KU [G] where G is a certain product of integral Eilenberg-Mac Lane spaces, and to a free commutative KU-algebra on a rational KU-module. We prove that S n ⊗KU is equivalent to KU [K(Z, n + 2)] and to F (Σ n KU Q). We describe the topological André-Quillen homology of KU as KU Q .

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“…Proof. We learned of results similar to the following from [Vee18]: if S → A → B are cofibrations of commutative S-algebras, then there is a weak equivalence…”

Section: σY ⊗ Kumentioning

confidence: 94%

Section: Iterated Topological Hochschild Homology Of Kumentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Higher topological Hochschild homology of periodic complex K-theory

Stonek

2020

Topology and its Applications

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“…Example 8.10. Veen's calculation [22] of the homotopy ring of the double THH of k, as k[µ ′ 2 , µ ′′ 2 ]⊗Λ(λ 3 ) gives an example whose coefficient ring is Noetherian and of Krull dimension 2.…”

Section: Amentioning

confidence: 99%

Ausoni–Bökstedt duality for topological Hochschild homology

Greenlees

2016

Journal of Pure and Applied Algebra

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Abstract. We consider the Gorenstein condition for topological Hochschild homology, and show that it holds remarkably often. More precisely, if R is a commutative ring spectrum and R −→ k is a map to a field of characteristic p then, provided k is small as an R-module, T HH(R; k) is Gorenstein in the sense of [11]. In particular, this holds if R is a (conventional) regular local ring with residue field k of characteristic p.Using only Bökstedt's calculation of T HH(k), this gives a non-calculational proof of dualities observed by Bökstedt [9]

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Algebraic 𝐾-theory of 𝑇𝐻𝐻(𝔽_{𝕡})

Bayındır¹,

Moulinos

2022

Trans. Amer. Math. Soc.

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In this work we study the E ∞ E_{\infty } -ring THH ⁡ ( F p ) \operatorname {THH}(\mathbb {F}_p) as a graded spectrum. Following an identification at the level of E 2 E_2 -algebras with F p [ Ω S 3 ] \mathbb {F}_p[\Omega S^3] , the group ring of the E 1 E_1 -group Ω S 3 \Omega S^3 over F p \mathbb {F}_p , we show that the grading on THH ⁡ ( F p ) \operatorname {THH}(\mathbb {F}_p) arises from decomposition on the cyclic bar construction of the pointed monoid Ω S 3 \Omega S^3 . This allows us to use trace methods to compute the algebraic K K -theory of THH ⁡ ( F p ) \operatorname {THH}(\mathbb {F}_p) . We also show that as an E 2 E_2 H F p H\mathbb {F}_p -ring, THH ⁡ ( F p ) \operatorname {THH}(\mathbb {F}_p) is uniquely determined by its homotopy groups. These results hold in fact for THH ⁡ ( k ) \operatorname {THH}(k) , where k k is any perfect field of characteristic p p . Along the way we expand on some of the methods used by Hesselholt-Madsen and later by Speirs to develop certain tools to study the THH of graded ring spectra and the algebraic K K -theory of formal DGAs.

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Detecting periodic elements in higher topological Hochschild homology

Abstract: Given a commutative ring spectrum R let Λ X R be the Loday functor constructed by Brun, Carlson and Dundas.

Cited by 14 publications

References 25 publications

Higher topological Hochschild homology of periodic complex K-theory

Higher topological Hochschild homology of periodic complex K-theory

Ausoni–Bökstedt duality for topological Hochschild homology

Algebraic 𝐾-theory of 𝑇𝐻𝐻(𝔽_{𝕡})

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