Maria Basterra, Michael A. Mandell
We describe an iterable construction of THH for an E_n ring spectrum. The reduced version is an iterable bar construction and its n-th iterate gives a model for the shifted cotangent complex at the augmentation, representing reduced topological Quillen homology of an augmented E_n algebra.
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We offer a complete description of $THH(E(2))$ under the assumption that the Johnson-Wilson spectrum $E(2)$ at a chosen odd prime carries an $E_\infty$-structure. We also place $THH(E(2))$ in a cofiber sequence $E(2) \rightarrow THH(E(2))\rightarrow \overline{THH}(E(2))$ and describe $\overline{THH}(E(2))$ under the assumption that $E(2)$ is an $E_3$-ring spectrum. We state general results about the $K(i)$-local behaviour of $THH(E(n))$ for all $n$ and $0 \leq i \leq n$. In particular, we compute $K(i)_*THH(E(n))$.
Motivated by a result from string topology, we prove a duality in topological Hochschild homology (THH). The duality relates the THH of an E_1-algebra spectrum and the THH of its derived Koszul dual algebra under certain compactness conditions. The result relies on results about module categories which may be of interest on their own. Finally, we relate this result to topological field theories and outline some future work.
Tensoring finite pointed simplicial sets with commutative ring spectra yields important homology theories such as (higher) topological Hochschild homology and torus homology. We prove several structural properties of these constructions relating $X \otimes (-)$ to $ΣX \otimes (-)$ and we establish splitting results. This allows us, among other important examples, to determine $THH^{[n]}_*(\mathbb{Z}/p^m; \mathbb{Z}/p)$ for all $n \geq 1$ and for all $m \geq 2$.
The topological Hochschild homology THH(R) of a commutative S-algebra (E_infty ring spectrum) R naturally has the structure of a commutative R-algebra in the strict sense, and of a Hopf algebra over R in the homotopy category. We show, under a flatness assumption, that this makes the Boekstedt spectral sequence converging to the mod p homology of THH(R) into a Hopf algebra spectral sequence. We then apply this additional structure to the study of some interesting examples, including the commutative S-algebras ku, ko, tmf, ju and j, and to calculate the homotopy groups of THH(ku) and THH(ko) af…
In this paper, we prove that both topological Hochschild homology and topological cyclic homology are sheaves for the fpqc topology on connective commutative ring spectra, by exploiting the May filtration on topological Hochschild homology.
We prove a conjecture of Hesselholt and Ausoni-Rognes, establishing localization cofiber sequences of spectra for THH(ku) and TC(ku). These sequences support Hesselholt's view of the map l to ku as a "tamely ramified" extension of ring spectra, and validate the hypotheses necessary for Ausoni's simplified computation of V(1)_* K(KU). In order to make sense of the relative term THH(ku|KU) in the cofiber sequence and prove these results, we develop a theory of THH and TC of Waldhausen categories and prove the analogues of Waldhausen's theorems for K-theory. We resolve the longstanding confusion …
This sketch argues that work of Hesselholt on the topological Hochschild homology of $\Cp$ extends, using work of Scholze and others, to define complex orientations for a version of topological Hochschild homology for rings of integers in a natural class of generalized cyclotomic perfectoid fields; and that the resulting spectra provide geometrically interesting targets for analogs of the Chern character, defined for certain integral lifts of the extraordinary $K$-functors of chromatic homotopy theory.
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 $THH$ 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.
We extend Torleif Veen's calculation of higher topological Hochschild homology ${\sf THH}^{[n]}_*(\mathbb{F}_p)$ from $n\leq 2p$ to $n\leq 2p+2$ for $p$ odd, and from $n=2$ to $n\leq 3$ for $p=2$. We calculate higher Hochschild homology ${\sf HH}_*^{[n]}(k[x])$ over $k$ for any integral domain $k$, and ${\sf HH}_*^{[n]}(\mathbb{F}_p[x]/x^{p^\ell})$ for all $n>0$. We use this and étale descent to calculate ${\sf HH}_*^{[n]}(\mathbb{F}_p[G])$ for all $n>0$ for any cyclic group $G$, and therefore also for any finitely generated abelian group $G$. We show a splitting result for higher ${\sf THH}$ …
Let $K(\mathbb{F}_q)$ be the algebraic $K$-theory spectrum of the finite field with $q$ elements and let $p \geq 5$ be a prime number coprime to $q$. In this paper we study the mod $p$ and $v_1$ topological Hochschild homology of $K(\mathbb{F}_q)$, denoted $V(1)_*THH(K(\mathbb{F}_q))$, as an $\mathbb{F}_p$-algebra. The computations are organized in four different cases, depending on the mod $p$ behaviour of the function $q^n-1$. We use different spectral sequences, in particular the Bökstedt spectral sequence and a generalization of a spectral sequence of Brun developed in an earlier paper. We…
Integral topological Hochschild homology of connective complex K-theory
本文通过motivic谱序列计算连通复数K理论的THH，并展示其在E_2页退化。
We describe the topological Hochschild homology of the periodic complex $K$-theory spectrum, $THH(KU)$, as a commutative $KU$-algebra: it is equivalent to $KU[K(\mathbb{Z},3)]$ and to $F(ΣKU_{\mathbb{Q}})$, where $F$ is the free commutative $KU$-algebra functor on a $KU$-module. Moreover, $F(ΣKU_{\mathbb{Q}})\simeq KU \vee ΣKU_{\mathbb{Q}}$, 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 $THH^n(KU)$, the $n$-fold iteration of $THH(KU)$, i.e. $T^n\otim…
We provide a new description of logarithmic topological André-Quillen homology in terms of the indecomposables of an augmented ring spectrum. The new description allows us to interpret logarithmic TAQ as an abstract cotangent complex, and leads to an base-change formula for logarithmic topological Hochschild homology. The latter is analogous to results of Weibel-Geller for Hochschild homology of discrete rings, and of McCarthy-Minasian and Mathew for topological Hochschild homology. For example, our results imply that logarithmic THH satisfies base-change for tamely ramified extensions of disc…
The algebraic structure of twisted topological Hochschild homology
本文研究扭曲THH的代数结构，计算了实bordism谱的$C_2$-扭曲THH，并扩展了其到扭曲Bökstedt谱序列，证明其为交换$\underline{E}_\star(R)$-代数。
Logarithmic topological Hochschild homology of the truncated Brown--Peterson spectrum
研究截断Brown-Peterson谱BP{n}的对数拓扑Hochschild同调，通过预对数结构和谱序列分析，计算了n=2时V(2)-同伦群。
We define a relative version of the Loday construction for a sequence of commutative S-algebras $A \rightarrow B \rightarrow C$ and a pointed simplicial subset $Y \subset X$. We use this to construct several spectral sequences for the calculation of higher topological Hochschild homology and apply those for calculations in some examples that could not be treated before.
We compute the $\mathbb{A}^1$-localization of several invariants of schemes namely, topological Hochschild homology ($\mathrm{THH}$), topological cyclic homology ($\mathrm{TC}$) and topological periodic cyclic homology ($\mathrm{TP}$). This procedure is quite brutal and kills the completed versions of most of these invariants. The main ingredient for the vanishing statements is the vanishing of $\mathbb{A}^1$-localization of de Rham cohomology (and, eventually, crystalline cohomology) in positive characteristics.
We study the C_p-equivariant Tate construction on the topological Hochschild homology THH(B) of a symmetric ring spectrum B by relating it to a topological version R_+(B) of the Singer construction, extended by a natural circle action. This enables us to prove that the fixed and homotopy fixed point spectra of THH(B) are p-adically equivalent for B = MU and BP. This generalizes the classical C_p-equivariant Segal conjecture, which corresponds to the case B = S.
In this work, we compute the topological coHochschild homology (coTHH) of interesting coalgebras such as the Steenrod algebra spectrum. For this, we start by extending the Hess-Shipley definition of coTHH to $\infty$-categories, following the Nikolaus-Scholze approach to THH. Furthermore, we prove that coTHH of what we call quasi-proper coalgebras can be obtained from THH via Spanier-Whitehead duality which provides further insight into coTHH and its relationship to THH.
We construct localization cofiber sequences for the topological Hochschild homology (THH) and topological cyclic homology (TC) of spectral categories. Using a global construction of the THH and TC of a scheme in terms of the perfect complexes in a spectrally enriched version of the category of unbounded complexes, the sequences specialize to localization cofiber sequences associated to the inclusion of an open subscheme. These are the targets of the cyclotomic trace from the localization sequence of Thomason-Trobaugh in K-theory. We also deduce versions of Thomason's blow-up formula and the pr…
We identify the topological Hochschild homology (THH) of the Thom spectrum associated to an E_\infty classifying map X -> BG, for G an appropriate group or monoid (e.g. U, O, and F). We deduce the comparison from the observation of McClure, Schwanzl, and Vogt that THH of a cofibrant commutative S-algebra (E_\infty ring spectrum) R can be described as an indexed colimit together with a verification that the Lewis-May operadic Thom spectrum functor preserves indexed colimits. We prove a splitting result THH(Mf) \htp Mf \sma BX_+ which yields a convenient description of THH(MU). This splitting ho…
We calculate the topological Hochschild homology groups of a maximal order in a central algebra over the rationals. Since the positive-dimensional THH groups consist only of torsion, we do this one prime ideal at a time for all the nonzero prime ideals in the center of the maximal order. This allows us to reduce the problem to studying the THH groups of maximal orders A in simple algebras over Q_p. We show that the topological Hochschild homology of A/(p) splits as the tensor product of its Hochschild homology and the topological Hochschild homology of F_p. We use this result in Brun's spectra…
A Note on Topological Hochschild Homology Relative to $\Sphere_{W(k)}[x_0,x_1,\ldots,x_n]$
本文研究了相对拓扑霍赫希尔德同调与尼加德完备化的弗罗贝尼乌斯扭化相对棱柱上同调的关系，并以$R=\text{Z}_p[x]/(px)$为例，通过下降方法计算了其$p$-完备化拓扑霍赫希尔德同调的同伦群。
We calculate the integral homotopy groups of THH(l) at any prime and of THH(ko) at p=2, where l is the Adams summand of the connective complex p-local K-theory spectrum and ko is the connective real K-theory spectrum.
Whitehead Filtrations for Computations in Topological Hochschild Homology
本研究利用Whitehead滤过导出的谱序列结合循环不变性，简化连通环谱的拓扑Hochschild同调计算，给出多个具体算例并推广至一般余纤维情形，还构建了2-局部自共轭K-理论对应计算的通用框架。
An HKR theorem for factorization homology
本文证明了一个HKR定理，用于计算光滑交换环谱的分解同调，修正并推广了McCarthy-Minasian的THH计算，并应用于有理KU的高阶THH。
$THH$ of the Morava $E$-theory Spectrum $E_{2}$
该研究通过Bökstedt谱序列计算$THH(E_{2})$的$K(i)$-同调并提升生成类，将$THH(E_{2})$描述为连续上纤维序列链的一部分，为利用迹方法研究$E_{n}$的代数$K$-理论提供基础。
In this paper we continue our study of logarithmic topological Hochschild homology. We show that the inclusion of the connective Adams summand into the p-local complex connective K-theory spectrum, equipped with suitable log structures, is a formally log THH-etale map, and compute the V(1)-homotopy of their logarithmic topological Hochschild homology spectra. As an application, we recover Ausoni's computation of the V(1)-homotopy of the ordinary THH of ku.
On endomorphisms of topological Hochschild homology
本文计算了稳定∞-范畴上拓扑Hochschild同调（THH）函子及其变体（包括HH_k和稳定对称幺半范畴上的THH）的自同态。
Let A be an A_\infty ring spectrum. We use the description from [2] of the cyclic bar and cobar construction to give a direct definition of topological Hochschild homology and cohomology of A using the Stasheff associahedra and another family of polyhedra called cyclohedra. This construction builds the maps making up the A_\infty structure into THH(A), and allows us to study how THH(A) varies over the moduli space of A_\infty structures on A. As an example, we study how topological Hochschild cohomology of Morava K-theory varies over the moduli space of A_\infty structures and show that in the…
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 and R----->k is a ring map to a field of characteristic p then, provided k is small as an R-module, THH(R;k) is Gorenstein in the sense of Dwyer-Greenlees-Iyengar. In particular, this holds if R is a (conventional) regular local ring with residue field k of characteristic p. Using only Bokstedt's calculation of THH(k), this gives a non-calculational proof of dualities observed in calculations by Bokstedt, McClure-Staffeldt, A…
We define a topological Hochschild (THH) and cyclic (TC) homology theory for differential graded (dg) categories and construct several non-trivial natural transformations from algebraic K-theory to THH(-). In an intermediate step, we prove that the homotopy theory of dg categories is Quillen equivalent, through a four step zig-zag of Quillen equivalences, to the homotopy theory of Eilenberg-Mac Lane spectral categories. Finally, we show that over the rationals two dg categories are topological equivalent if and only if they are quasi-equivalent.
Equivariant Witt Complexes and Twisted Topological Hochschild Homology
本文研究扭曲拓扑霍奇同调（THH）的等变同伦群代数结构，定义等变Witt复形并证明其捕捉扭曲THH等变同伦的结构，为等变代数研究提供新贡献。
Extension DGAs and topological Hochschild homology
本文研究由环谱经标量扩张得到的扩展DGAs，证明其THH谱可分裂，且形式DGAs的THH群可由Hochschild同调得到，并探讨了拓扑等价与拟同构的一致性。
We show that the topological Hochschild homology THH(R of an E_n-ring spectrum R is an E_{n-1}-ring spectrum. The proof is based on the fact that the tensor product of the operad Ass for monoid structures and the the little n-cubes operad is an E_{n+1}-operad, a result which is of independent interest.
On the Residue Sequence in Logarithmic Topological Cyclic Homology
本研究通过关联循环谱局部化序列与对数THH留数序列解决Rognes猜想，证明对数THH、TR、TC是相关构造的严格推广且无需正则假设即可有局部化序列，结合已有工作证明存在稳定无穷范畴实现特定对数结构的对数项，并提出该范畴可反映额外结构的猜想及离散赋值环情形的证据。
We specify exterior generators for $π_* THH(MU) = π_*(MU) \otimes E(λ'_n \mid n\ge1)$ and $π_* THH(BP) = π_*(BP) \otimes E(λ_n \mid n\ge1)$, and calculate the action of the $σ$-operator on these graded rings. In particular, $σ(λ'_n) = 0$ and $σ(λ_n) = 0$, while the actions on $π_*(MU)$ and $π_*(BP)$ are expressed in terms of the right units $η_R$ in the Hopf algebroids $(π_*(MU), π_*(MU \wedge MU))$ and $(π_*(BP), π_*(BP \wedge BP))$, respectively.
The singular chain complex of the iterated loop space is expressed in terms of the cobar construction. After that we consider the spectral sequence of the cobar construction and calculate its first term over Z/p-coefficients and over a field of characteristic zero. Finally we apply these results to calculate the homology of the iterated loop spaces of the stunted real and complex projective spaces. In the Appendix, written by F.Sergeraert there are considered computer methods for calculations of the homology of iterated loop spaces.
We determine higher topological Hochschild homology of rings of integers in number fields with coefficients in suitable residue fields. We use the iterative description of higher THH for this and Postnikov arguments that allow us to reduce the necessary computations to calculations in homological algebra, starting from the results of Bökstedt and Lindenstrauss-Madsen on (ordinary) topological Hochschild homology.
The first author proved in a previous paper that the n-fold bar construction for commutative algebras can be generalized to E_n-algebras, and that one can calculate E_n-homology with trivial coefficients via this iterated bar construction. We extend this result to E_n-homology and E_n-cohomology of a commutative algebra A with coefficients in a symmetric A-bimodule.
We show the topological Hochschild homology spectrum of a twisted group algebra $\THH(A^τ[G])$ is the Thom spectrum associated to a parametrized orthogonal spectrum $E(A,G)$. We then analyze the structure of the parametrized orthogonal spectrum $E(A,G)$ and show that it is locally trivial.
Let $Q$ denote MacLane's $Q$-construction, and $\otimes$ denote the smash product of spectra. In this paper we construct an equivalence $Q(R)\simeq \mathbb Z\otimes R$ in the category of $A_\infty$ ring spectra for any ring $R$, thus proving a conjecture made by Fiedorowicz, Schwänzl, Vogt and Waldhausen in "MacLane homology and topological Hochschild homology". More precisely, we construct is a symmetric monoidal structure on $Q$ (in the $\infty$-categorical sense) extending the usual monoidal structure, for which we prove an equivalence $Q(-)\simeq \mathbb Z\otimes -$ as symmetric monoidal f…
If X is a cosimplical $E_{n+1}$ space then Tot(X) is an $E_{n+1}$ space and its mod 2 homology $H_*(Tot(X))$ has Dyer-Lashof and Browder operations. It's natural to ask if the spectral sequence converging to $H_*(Tot(X))$ admits compatible operations. In this paper I give a positive answer to this question.
Topological Hochschild homology of truncated Brown-Peterson spectra I
本文计算了带系数的截断Brown-Peterson谱充分结构化形式的拓扑Hochschild同调，涵盖特定素数下$BP\langle n\rangle$的$E_3$形式及Hill-Lawson构造的$E_\infty$形式，提供了完整的THH计算结果。
We proved in a previous article that the bar complex of an E-infinity algebra inherits a natural E-infinity algebra structure. As a consequence, a well-defined iterated bar construction B^n(A) can be associated to any algebra over an E-infinity operad. In the case of a commutative algebra A, our iterated bar construction reduces to the standard iterated bar complex of A. The first purpose of this paper is to give a direct effective definition of the iterated bar complexes of E-infinity algebras. We use this effective definition to prove that the n-fold bar complex B^n(A) admits an extension to…
Let GL_1(R) be the units of a commutative ring spectrum R. In this paper we identify the composition BGL_1(R)->K(R)->THH(R)->Ω^{\infty}(R), where K(R) is the algebraic K-theory and THH(R) the topological Hochschild homology of R. As a corollary we show that classes in π_{i-1}(R) not annihilated by the stable Hopf map give rise to non-trivial classes in K_i(R) for i\geq 3.
We define twisted Hochschild homology for Green functors. This construction is the algebraic analogue of the relative topological Hochschild homology $THH_{C_n}(-)$, and it describes the $E_2$ term of the Künneth spectral sequence for relative $THH$. Applied to ordinary rings, we obtain new algebraic invariants. Extending Hesselholt's construction of the Witt vectors of noncommutative rings, we interpret our construction as providing Witt vectors for Green functors.
Manfred Stelzer has pointed out that part of Corollary 4.5 of our paper "Topological nonrealization results via the Goodwillie tower approach to iterated loopspace homology" [Alg. Geo. Top. 8 (2008), 2109--2129] was not sufficiently proved, and, indeed, is likely incorrect as stated. This necessitates a little more argument to finish the proof of the main theorem of the original paper. The statement of this theorem, and all the examples, remain unchanged.
We prove a strengthened version of a theorem of Lionel Schwartz that says that certain modules over the Steenrod algebra cannot be the mod 2 cohomology of a space. What is most interesting is our method, which replaces his iterated use of the Eilenberg--Moore spectral sequence by a single use of the spectral sequence converging to the mod 2 cohomology of Omega^nX obtained from the Goodwillie tower for the suspension spectrum of Omega^nX. Much of the paper develops basic properties of this spectral sequence.
We define a functorial spectrum-level filtration on the topological Hochschild homology of any commutative ring spectrum $R$, and more generally the factorization homology $R \otimes X$ for any space $X$, echoing algebraic constructions of Loday and Pirashvili. We investigate the properties of this filtration and show that it breaks THH up into common eigenspectra of the Adams operations.
A rational model for the fiberwise THH transfer I: Sullivan algebras
本文证明纤维化THH转移可由Sullivan模型的Hochschild同调转移理性建模，通过高阶范畴追踪理论和Braunack-Mayer的工作实现。
Topological Hochschild homology of the image of j
本文计算了j像谱的THH并证明了Segal猜想，揭示了THH与p进圆自由循环空间的关系。
Iterated Topological Hochschild Homology of $\mathbb{F}_p$ at odd primes
本文计算奇数素数下 $\mathbb{F}_p$ 的迭代拓扑 Hochschild 同调的同伦群，通过 bar 谱序列证明其在 $E^2$ 页坍缩，并确定环结构。
We flesh out the theory of "trace theories" and "trace functors" sketched in arXiv:1308.3743, extend it to a homotopical setting, and prove a reconstruction theorem claiming that a trace theory is completely determined by the associated trace functor. As an application, we consider Topological Hoshschild Homology $THH(A,M)$ of a algebra $A$ over a perfect field of positive characteristic, with coefficients in a bimodule $M$, and prove two comparison results. Firstly, we give a very simple algebraic model for THH in terms of Hochschild-Witt Homology WHH of arXiv:1604.01588 (and we also identify…
In this paper we present a new approach to computing homology (with field coefficients) and persistent homology. We use concepts from discrete Morse theory, to provide an algorithm which can be expressed solely in terms of simple graph theoretical operations. We use iterated Morse decomposition, which allows us to sidetrack many problems related to the standard discrete Morse theory. In particular, this approach is provably correct in any dimension.
In this article, we study the topological Hochschild homology of $\mathbf{E}_3$-forms of truncated Brown-Peterson spectra, taken relative to certain Thom spectra $X(p^n)$ (introduced by Ravenel and used by Devinatz-Hopkins-Smith in the proof of the nilpotence theorem). We prove analogues of Bökstedt's calculations $\mathrm{THH}(\mathbf{F}_p) \simeq \mathbf{F}_p[ΩS^3]$ and $\mathrm{THH}(\mathbf{Z}_p) \simeq \mathbf{Z}_p[ΩS^3\langle{3}\rangle]$. We also construct a topological analogue of the Sen operator of Bhatt-Lurie-Drinfeld, and study a higher chromatic extension. The behavior of these "top…
Real Topological Hochschild Homology of Perfectoid Rings
本文将Bhatt-Morrow-Scholze关于THH的结果推广到THR，计算了完美胚环的THR，并给出HKR定理的实精细化，为在准辛同伦环上建立动机滤过提供工具。
Twisted Bicategorical Shadows and Traces
本文引入带$G$-扭曲数据的双范畴，构造$g$-扭曲双范畴影子与迹，证明其$G$-Morita不变性，可恢复$g$-扭曲Hattori–Stallings迹与零次扭曲Dennis迹，适用于$C_n$-Green函子等对象。
Twisted topological Hochschild homology of $C_n$-equivariant spectra was introduced by Angeltveit, Blumberg, Gerhardt, Hill, Lawson, and Mandell, building on the work of Hill, Hopkins, and Ravenel on norms in equivariant homotopy theory. In this paper we introduce tools for computing twisted THH, which we apply to computations for Thom spectra, Eilenberg-MacLane spectra, and the real bordism spectrum $MU_{\mathbb{R}}$. In particular, we construct an equivariant version of the Bökstedt spectral sequence, the formulation of which requires further development of the Hochschild homology of Green f…
We introduce notions of unramified and totally ramified maps in great generality - for commutative rings, schemes, ring spectra, or derived schemes. We prove that the definition is equivalent to the classical definition in the case of rings of integers in number fields. The new definition leads directly (without computational techniques) to a calculation of topological Hochschild homology for rings of integers. We show that THH(R) is the homotopy cofiber of a map $R[ΩS^3]\otimes_RΩ^1_{R/\mathbb{Z}}\to R[ΩS^3\langle 3\rangle]$, so there is a long exact sequence $\cdots\to H_\ast(ΩS^3;Ω^1_{R/\ma…
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