In this paper we analyze the higher topological Hochschild homology of commutative Thom S-algebras. This includes the case of the classical cobordism spectra MO, MSO, MU, etc. We consider the homotopy orbits of the torus action on iterated topological Hochschild homology and we describe the relationship to topological Andre-Quillen homology.
We present a study of the homological algebra of bimodules over $A_\infty$-algebras endowed with an involution. Furthermore we introduce a derived description of Hochschild homology and cohomology for involutive $A_\infty$-algebras.
We compute the component group of the derived $G$-geometric fixed points of the real topological Hochschild homology of a ring with anti-involution, where $G$ denotes the Galois group Gal($\mathbb{C}$/$\mathbb{R}$) of order 2.
We generalize a spectral sequence of Brun for the computation of topological Hochschild homology. The generalized version computes the $E$-homology of $THH(A;B)$, where $E$ is a ring spectrum, $A$ is a commutative $S$-algebra and $B$ is a connective commutative $A$-algebra. The input of the spectral sequence are the topological Hochschild homology groups of $B$ with coefficients in the $E$-homology groups of $B \wedge_A B$. The mod $p$ and $v_1$ topological Hochschild homology of connective complex $K$-theory has been computed by Ausoni and later again by Rognes, Sagave and Schlichtkrull. We p…
We develop a theory of R-module Thom spectra for a commutative symmetric ring spectrum R and we analyze their multiplicative properties. As an interesting source of examples, we show that R-algebra Thom spectra associated to the special unitary groups can be described in terms of quotient constructions on R. We apply the general theory to obtain a description of the R-based topological Hochschild homology associated to an R-algebra Thom spectrum.
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.
In this short note we study the topological Hoschschild homology of Eilenberg-MacLane spectra for finite cyclic groups. In particular, we show that the Eilenberg-MacLane spectrum H(Z/p^k) is a Thom spectrum for any prime p (except, possibly, when p=k=2) and we also compute its topological Hoschshild homology. This yields a short proof of the results obtained by Brun, and by Pirashvili except for the anomalous case p=k=2.
Let $A$ be a commutative algebra over the field ${\mathbb F}_2 = {\mathbb Z}/2$. We show that there is a natural algebra homomorphism $\ell (A) \to HC^-_*(A)$ which is an isomorphism when $A$ is a smooth algebra. Thus, the functor $\ell$ can be viewed as an approximation of negative cyclic homology and ordinary cyclic homology $HC_*(A)$ is a natural $\ell (A)$-module. In general, there is a spectral sequence $E^2 = L_*(\ell )(A) \Rightarrow HC_*^- (A)$. We find associated approximation functors $\ell^+$ and $\ell^{per}$ for ordinary cyclic homology and periodic cyclic homology, and set up thei…
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 give a new proof, independent of Lin's theorem, of the Segal conjecture for the cyclic group of order two. The key input is a calculation, as a Hopf algebroid, of the Real topological Hochschild homology of $\mathbb{F}_2$. This determines the $\mathrm{E}_2$-page of the descent spectral sequence for the map $\mathrm{N}\mathbb{F}_2 \to \mathbb{F}_2$, where $\mathrm{N}\mathbb{F}_2$ is the $C_2$-equivariant Hill--Hopkins--Ravenel norm of $\mathbb{F}_2$. The $\mathrm{E}_2$-page represents a new upper bound on the $RO(C_2)$-graded homotopy of $\mathrm{N}\mathbb{F}_2$, from which the Segal conject…
We show that Ravenel's spectrum $X(2)$ is the versal $E_1$-$S$-algebra of characteristic $η$. This implies that every $E_1$-$S$-algebra $R$ of characteristic $η$ admits an $E_1$-ring map $X(2)\to R$, i.e. an $\mathbb{A}_\infty$ complex orientation of degree 2. This implies that $R^\ast(\mathbb{C}P^2)\cong R_\ast[x]/x^3$. Additionally, if $R$ is an $\mathbb{E}_2$-ring Thom spectrum admitting a map (of homotopy ring spectra) from $X(2)$, e.g. $X(n)$, its topological Hochschild homology has a simple description.
We give a description of the factorization homology and $E_n$ topological Hochschild cohomology of Thom spectra arising from $n$-fold loop maps $f: A \to BO$, where $A = Ω^n X$ is an $n$-fold loop space. We describe the factorization homology $\int_M Th(f)$ as the Thom spectrum associated to a certain map $\int _M A \to BO$. When $M$ is framed and $X$ is $(n-1)$-connected, this spectrum is equivalent to a Thom spectrum of a virtual bundle over the mapping space $Map_c(M,X)$; in general, this is a Thom spectrum of a virtual bundle over a certain section space. This can be viewed as a twisted fo…
We compute topological Hochschild homology mod $p$ and $v_1$ of the connective cover of the $K(1)$-local sphere spectrum for all primes $p\ge 3$. This is accomplished using a May-type spectral sequence in topological Hochschild homology constructed from a filtration of a commutative ring spectrum.
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…
Hochschild (co)homology and Pirashvili's higher order Hochschild (co)homology are useful tools for a variety of applications including deformations of algebras. When working with higher order Hochschild (co)homology, we can consider the (co)homology of any commutative algebra with symmetric coefficient bimodules, however traditional Hochschild (co)homology is able to be computed for any associative algebra with not necessarily symmetric coefficient bimodules. In a previous paper, the author generalized higher order Hochschild cohomology for multimodule coefficients (which need not be symmetric…
We show that an important classical fixed point invariant, the Reidemeister trace, arises as a topological Hochschild homology transfer. This generalizes a corresponding classical result for the Euler characteristic and is a first step in showing the Reidemeister trace is in the image of the cyclotomic trace. The main result follows from developing the relationship between shadows, topological Hochschild homology, and Morita invariance in bicategorical generality.
Higher Hochschild homology is the analog of the homology of spaces, where the context for the coefficients -- which usually is that of abelian groups -- is that of commutative algebras. Two spaces that are equivalent after a suspension have the same homology. We show that this is not the case for higher Hochschild homology, providing a counterexample to a behavior so far observed in stable homotopy theory.
Shadows for bicategories, defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. In this paper, we define Hochschild-type invariants for monoids in a symmetric monoidal, simplicial model category $\mathsf V$, as well as for small $\mathsf V$-categories. We show that each of these constructions extends to a shadow on an appropriate bicategory, which implies in particular that they are Morita invariant. We also define a generalized theory of Hochschild homology twisted by an automorphism and show that it is Morita invariant. Hochschild homolo…
We perform Hochschild homology calculations in the algebro-geometric setting of motives. The motivic Hochschild homology coefficient ring contains torsion classes which arise from the mod-$p$ motivic Steenrod algebra and from generating functions on the natural numbers with finite non-empty support. Under the Betti realization, we recover Bökstedt's calculation of the topological Hochschild homology of finite prime fields.
We study the homological algebra of bimodules over involutive associative algebras. We show that Braun's definition of involutive Hochschild cohomology in terms of the complex of involution-preserving derivations is indeed computing a derived functor: the Z/2-invariants intersected with the center. We then introduce the corresponding involutive Hochschild homology theory and describe it as the derived functor of the pushout of Z/2-coinvariants and abelianization.
We show that Connes' B-operator on a cyclic differential graded k-module M is a model for the canonical circle action on the geometric realization of M. This implies that the negative cyclic homology and the periodic cyclic homology of a differential graded category can be identified with the homotopy fixed points and the Tate fixed points of the circle action on its Hochschild complex.
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.
We show that the Hochschild-Pirashvili homology on any suspension admits the so called Hodge splitting. For a map between suspensions $f\colon ΣY\to ΣZ$, the induced map in the Hochschild-Pirashvili homology preserves this splitting if $f$ is a suspension. If $f$ is not a suspension, we show that the splitting is preserved only as a filtration. As a special case, we obtain that the Hochschild-Pirashvili homology on wedges of circles produces new representations of $Out(F_n)$ that do not factor in general through $GL(n,Z)$. The obtained representations are naturally filtered in such a way that …
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 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…
