Twisted Hochschild homology and MacLane homology

On the Hochschild homology of smash biproducts

We develop a new spectral sequence in order to calculate Hochschild homology of smash biproducts (also called twisted tensor products) of unital associative algebras $A\# B$ provided one of $A$ or $B$ has Hochschild dimension less than 2. We use this spectral sequence to calculate Hochschild homology of quantum tori, multiparametric quantum affine spaces, quantum complete intersections, quantum Weyl algebras, deformed completed Weyl algebras, and finally the algebra $M_q(2)$ of quantum $2\times 2$-matrices.

关于Eilenberg-MacLane谱的拓扑André-Quillen同调

On Topological André-Quillen homology of Eilenberg-MacLane spectra

本文基于Dundas等人工作，计算了Eilenberg-MacLane谱（如Hℤ和Hℤ/p^n）的拓扑André-Quillen同调，方法类似Cartan计算Steenrod代数，且更简短。

Jacobi-Zariski Exact Sequence for Hochschild Homology and Cyclic (Co)Homology

We prove that for an inclusion of unital associative but not necessarily commutative algebras $B\subseteq A$ we have long exact sequences in Hochschild homology and cyclic (co)homology akin to the Jacobi-Zariski sequence in André-Quillen homology, provided that the quotient $B$-bimodule $A/B$ is flat. We also prove that for an arbitrary r-flat morphism $f:B\to A$ with an H-unital kernel, one can express the Wodzicki excision sequence and the corresponding Jacobi-Zariski sequence in Hochschild homology and cyclic (co)homology as a single long exact sequence.

Excision in Hochschild and cyclic homology without continuous linear sections

We prove that continuous Hochschild and cyclic homology satisfy excision for extensions of nuclear H-unital Frechet algebras and use this to compute them for the algebra of Whitney functions on an arbitrary closed subset of a smooth manifold. Using a similar excision result for periodic cyclic homology, we also compute the periodic cyclic homology of algebras of smooth functions and Whitney functions on closed subsets of smooth manifolds.

Hochschild (co)homology and Koszul duality

In this article we discuss two different but related results on Hochschild (co)homology and the theory of Koszul duality. On the one hand, we prove essentially that the Tamarkin-Tsygan calculus of an Adams connected augmented dg algebra and of its Koszul dual are dual. This uses the fact that Hochschild cohomology and homology may be regarded as a twisted construction of some natural (augmented) dg algebras and dg modules over the former. In particular, from these constructions it follows that the computation of the cup product on Hochschild cohomology and cap product on Hochschild homology of…

弯曲代数的 Hochschild 同调

On the Hochschild Homology of Curved Algebras

本文计算了合适弯曲环上完美弯曲模微分分次范畴的 Hochschild 同调，提出“de Rham 模型”，推广了矩阵分解的先前结果，并在附录中包含了作为证明关键的 Briggs 定理的证明。

Topological Hochschild homology and the Bass trace conjecture

We use the methods of topological Hochschild homology to shed new light on the groups satisfying the Bass trace conjecture. We show that the factorization of the Hattori-Stallings rank map through the Bokstedt-Hsiang-Madsen cyclotomic trace map leads to Linnell's restriction on such groups. As a new consequence of this restriction, we show that the conjecture holds for any group G with the property that every subgroup that is isomorphic to the additive group of rational numbers has nontrivial and central image in some quotient of G.

Hochschild homology and cohomology of generalized Weyl algebras

We compute Hochschild homology and cohomology of a class of generalized Weyl algebras (for short GWA, defined by Bavula in St.Petersbourg Math. Journal 1999 4(1) pp. 71-90). Examples of such algebras are the n-th Weyl algebras, U(sl_2), primitive quotients of U(sl_2), and subalgebras of invariants of these algebras under finite cyclic groups of automorphisms. We answer a question of Bavula - Jordan (Trans. A.M.S. 353 (2) 2001 pp. 769 -794) concerning the generator of the group of automorphism of a GWA. We also explain previous results on the invariants of Weyl algebras and of primitive quotien…

Hochschild and cyclic homology of the crossed product of algebraic irrational rotational algebra by finite subgraoups of $SL(2,\mathbb Z)$

Let $Γ\subset SL(2, \mathbb Z)$ be a finite subgroup acting on the irrational rotational algebra $\mathcal A_θ$ via the restriction of the canonical action of $SL(2,\mathbb Z)$. Consider the crossed product algebra $\mathcal A_θ^{alg} \rtimes Γ$ obtained by the restriction of the $Γ$ action on the algberaic irrational rotational algebra. In this paper we prove many results on the homology group of the crossed product algebra $\mathcal A_θ^{alg} \rtimes Γ$. We also analyse the case of the smooth crossed product algebra, $\mathcal A_θ\rtimes Γ$ and calculate some of its homology groups.

Hochschild and cyclic (co)homology of superadditive categories

We define the Hochschild and cyclic (co)homology groups for superadditive categories and show that these (co)homology groups are graded Morita invariants. We also show that the Hochschild and cyclic homology are compatible with the tensor product of superadditive categories.

Hopf--Hochschild (co)homology of module algebras

We define a version of Hochschild homology and cohomology suitable for a class of algebras admitting compatible actions of bialgebras, called module algebras. We show this (co)homology, called Hopf--Hochschild (co)homology, can also be defined as a derived functor on the category of representations of a crossed product algebra. We investigate the relationship of our theory with Hopf cyclic cohomology and also prove Morita invariance of the Hopf--Hochschild (co)homology.

Tate-Hochschild homology and cohomology of Frobenius algebras

We study Tate-Hochschild homology and cohomology for a two-sided Noetherian Gorenstein algebra. These (co)homology groups are defined for all degrees, non-negative as well as negative, and they agree with the usual Hochschild (co)homology groups for all degrees larger than the injective dimension of the algebra. We prove certain duality theorems relating the Tate-Hochschild (co)homology groups in positive degree to those in negative degree, in the case where the algebra is Frobenius. We explicitly compute all Tate-Hochschild (co)homology groups for certain classes of Frobenius algebras, namely…

Two classes of algebras with infinite Hochschild homology

We prove without any assumption on the ground field that higher Hochschild homology groups do not vanish for two large classes of algebras whose global dimension in not finite.

Symplectic homology as Hochschild homology

These are (heavily revised) notes from lectures given at the AMS Algebraic Geometry meeting in Seattle, 2005. The main topic is symplectic homology seen from the point of view of Lefschetz fibrations. Most of the content is speculative, but some supporting evidence is included.

When the theories meet: Khovanov homology as Hochschild homology of links

We show that Khovanov homology and Hochschild homology theories share common structure. In fact they overlap: Khovanov homology of a $(2,n)$-torus link can be interpreted as a Hochschild homology of the algebra underlining the Khovanov homology. In the classical case of Khovanov homology we prove the concrete connection. In the general case of Khovanov-Rozansky, $sl(n)$, homology and their deformations we conjecture the connection. The best framework to explore our ideas is to use a comultiplication-free version of Khovanov homology for graphs developed by L. Helme-Guizon and Y. Rong and exten…

拓扑 Hochschild 同调中的等变运算

Equivariant operations in topological Hochschild homology

本文发现拓扑 Hochschild 同调与上同调之间的新等变关系，并通过计算有限素域的拓扑 Hochschild 上同调的拓扑 Hochschild 同调，将其解释为结构化运算环。

Hochschild homology of iterate skew polynomial rings

We study the Hochschild homology of the iterated skew polynomial rings introduced by D. Jordan in ``A simple localization of the quantized Weyl algebra''. First, we obtain a complex, smaller than the canonical one of Hochschild, given the homology of such an algebra, and then, we study this complex in order to compute the homology of some families of algebras. In particular we compute the homology of some quantum groups, in the generic case.

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

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.

On the Hochschild (co)homology of quantum exterior algebras

We compute the Hochschild cohomology and homology of a class of quantum exterior algebras, with coefficients in twisted bimodules. As a result we obtain several interesting examples of the homological behavior of these algebras.

Computational tools for twisted topological Hochschild homology of equivariant spectra

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…

Hochschild and cyclic (co)homology of the Fomin-Kirillov algebra on 3 generators

The goal of this article is to explicitly compute the Hochschild (co)homology of the Fomin-Kirillov algebra on three generators over a field of characteristic different from 2 and 3. We also obtain the cyclic (co)homology of the Fomin-Kirillov algebra in case the characteristic of the field is zero. Moreover, we compute the algebra structure of the Hochschild cohomology.

Hochschild homology and global dimension

We prove that for certain classes of graded algebras (Koszul, local, cellular), infinite global dimension implies that Hochschild homology does not vanish in high degrees, provided the characteristic of the ground field is zero. Our proof uses Igusa's formula relating the Euler characteristic of relative cyclic homology to the graded Cartan determinant.

The Dennis' Supertrace and the Hochschild Homology of Supermatrices

We construct, in this paper, a generalization of the Dennis trace (for matrices) to the case of the supermatrices over an arbitrary (not necessarily commutative) superalgebra with unit. By analogy with the ungraded case, we show how it is possible to use this map to construct an isomorphism from the Hochschild homology of the superalgebra to the Hochschild homology of the supermatrix algebra.

Topological Hochschild homology of twisted group algebras

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.

Hochschild (co)homology of the Dunkl operator quantization of $\Z_2$-singularity

We study Hochschild (co)homology groups of the Dunkl operator quantization of $\Z_2$-singularity constructed by Halbout and Tang. Further, we study traces on this algebra and prove a local algebraic index formula.

On the Hochschild (co)homology of Quantum Homogeneous Spaces

The recent result of Brown and Zhang establishing Poincare duality in the Hochschild (co)homology of a large class of Hopf algebras is extended to right coideal subalgebras over which the Hopf algebra is faithfully flat, and applied to the standard Podles quantum 2-sphere.

Maximal commutative subalgebras, Poisson geometry and Hochschild homology

A Poisson geometry arising from maximal commutative subalgebras is studied. A spectral sequence convergent to Hochschild homology with coefficients in a bimodule is presented. It depends on the choice of a maximal commutative subalgebra inducing appropriate filtrations. Its E^{2}_{p,q}-groups are computed in terms of canonical homology with values in a Poisson module defined by a given bimodule and a maximal commutative subalgebra.

Twisted Hochschild Homology of Quantum Flag Manifolds and Kähler Forms

We study the twisted Hochschild homology of quantum flag manifolds, the twist being the modular automorphism of the Haar state. We prove that every quantum flag manifold admits a non-trivial class in degree two, with an explicit representative defined in terms of a certain projection. The corresponding classical two-form, via the Hochschild-Kostant-Rosenberg theorem, is identified with a Kähler form on the flag manifold.

Localization sequences for logarithmic topological Hochschild homology

We study the logarithmic topological Hochschild homology of ring spectra with logarithmic structures and establish localization sequences for this theory. Our results apply, for example, to connective covers of periodic ring spectra like real and complex topological K-theory.

A shadow perspective on equivariant Hochschild homologies

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…

Derived invariance of Hochschild-Mitchell (co)homology and one-point extensions

In this article we prove derived invariance of Hochschild-Mitchell homology and cohomology and we extend to $k$-linear categories a result by Barot and Lenzing concerning derived equivalences and one-point extensions. We also prove the existence of a Happel long exact sequence and we give a generalization of this result which provides an alternative approach.

Higher Order Hochschild (Co)homology of Noncommutative Algebras

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…

The homotopy fixed points of the circle action on Hochschild homology

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.

On the Hochschild homology of l^1-rapid decay group algebras

We show that for many semi-hyperbolic groups the decomposition into conjugacy classes of the Hochschild homology of the l^1-rapid decay group algebra is injective.

Higher Hochschild homology is not a stable invariant

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.

Hochschild (co)homology of Hopf crossed products

Let A a k-algebra, H a Hopf algebra, E = A#H a general crossed product and M an E-bimodule. We obtain a complex simpler than the canonical one, giving the Hochschild homology of E with coefficients in M. This complex is eqquiped with a natural filtration. We prove that the associated spectral sequence coincides with that obtained by either, the Hochschild-Serre direct method or the Cartan-Leray-Grothendieck method. We also get similar results for the cohomology.

Hochschild homology of mod-$p$ motivic cohomology over algebraically closed fields

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.

Logarithmic topological Hochschild homology of topological K-theory spectra

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.

Hochschild homology and cohomology of down-up algebras

We present a detailed computation of the cyclic and the Hochschild homology and cohomology of generic and 3-Calabi-Yau homogeneous down-up algebras. This family was defined by Benkart and Roby in their study of differential posets. Our calculations are completely explicit, by making use of the Koszul bimodule resolution and arguments similar to those used by the second and third authors for Yang-Mills algebras.

Topological Hochschild homology and the cyclic bar construction in symmetric spectra

The cyclic bar construction in symmetric spectra and Bökstedt's original construction are two possible ways to define the topological Hochschild homology of a symmetric ring spectrum. In this short note we explain how to correct an error in Shipley's original comparison of these two approaches.

The homology of digraphs as a generalisation of Hochschild homology

J. Przytycki has established a connection between the Hochschild homology of an algebra $A$ and the chromatic graph homology of a polygon graph with coefficients in $A$. In general the chromatic graph homology is not defined in the case where the coefficient ring is a non-commutative algebra. In this paper we define a new homology theory for directed graphs which takes coefficients in an arbitrary $A-A$ bimodule, for $A$ possibly non-commutative, which on polygons agrees with Hochschild homology through a range of dimensions.

对数Hochschild同调的Hochschild-Kostant-Rosenberg定理与留数序列

A Hochschild-Kostant-Rosenberg theorem and residue sequences for logarithmic Hochschild homology

本文将Hochschild同调纳入对数动机研究框架，定义导出前对数环的对数Hochschild同调，构造André-Quillen型谱序列，证明对数Hochschild-Kostant-Rosenberg定理及该同调在对数动机范畴可表示，还推导出涉及对数概形爆破的广义留数序列。

实拓扑霍赫希尔德同调的计算工具

Computational tools for Real topological Hochschild homology

本文构建了收敛于实拓扑霍赫希尔德同调$\text{THR}$等变同调的实等变Bökstedt谱序列，并证明交换$C_2$-环谱$A$的$\text{THR}(A)$具有$C_2$-等变稳定同伦范畴下的$A$-Hopf algebroid结构，为实拓扑霍赫希尔德同调提供计算工具。

Localization theorems in topological Hochschild homology and topological cyclic homology

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…

一个朝向动机Hochschild同调的动机Greenlees谱序列

A motivic Greenlees spectral sequence towards motivic Hochschild homology

本文定义动机Greenlees谱序列，研究动机上同调谱模素数p的动机拓扑Hochschild同调，并在代数闭域特征与p互素时确定相关谱的同伦环。

Deleting or adding arrows of a bound quiver algebra and Hochschild (co)homology

We describe how the Hochschild (co)homology of a bound quiver algebra changes when adding or deleting arrows to the quiver. The main tools are relative Hochschild (co)homology, the Jacobi-Zariski long exact sequence obtained by A. Kaygun and a one step relative projective resolution of a tensor algebra.

Topological Hochschild homology of maximal orders in simple Q-algebras

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…

Simplicial Structures Over the 3-Sphere and Generalized Higher Order Hochschild Homology

In this paper we investigate the simplicial structure of a chain complex associated to the higher order Hochschild homology over the $3$-sphere. We also introduce the tertiary Hochschild homology corresponding to a quintuple $(A,B,C,\varepsilon,θ)$, which becomes natural after we organize the elements in a convenient manner. We establish these results by way of a bar-like resolution in the context of simplicial modules. Finally, we generalize the higher order Hochschild homology over a trio of simplicial sets, which also grants natural geometric realizations.

关于实Hochschild同调的滤过Hochschild-Kostant-Rosenberg定理

A filtered Hochschild-Kostant-Rosenberg theorem for real Hochschild homology

本文在$ C_2 $-Mackey函子中引入导出对合代数，证明其实Hochschild同调存在滤过且相关分次等同对合de Rham复形，并特化为Raksit的HKR-滤过Hochschild同调。

带有界极点的Hochschild和循环同调

Hochschild and cyclic Homologies with bounded poles

本文通过为模对构造Hochschild与循环同调，证明了其在带模动机范畴中的可表示性，并建立了模版HKR定理及模循环同调的可表示性。

A spectrum-level Hodge filtration on topological Hochschild homology

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(1) 的 Hochschild 同调与上同调

The Hochschild homology and cohomology of A(1)

本文利用若干 May 型谱序列，计算了由 $Sq^1$、$Sq^2$ 生成的 2-素 Steenrod 代数子代数 $A(1)$ 的 Hochschild 同调与上同调。

Comparing cyclotomic structures on different models for topological Hochschild homology

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.

A May-type spectral sequence for higher topological Hochschild homology

Given a filtration of a commutative monoid $A$ in a symmetric monoidal stable model category $\mathcal{C}$, we construct a spectral sequence analogous to the May spectral sequence whose input is the higher order topological Hochschild homology of the associated graded commutative monoid of $A$, and whose output is the higher order topological Hochschild homology of $A$. We then construct examples of such filtrations and derive some consequences: for example, given a connective commutative graded ring $R$, we get an upper bound on the size of the $THH$-groups of $E_{\infty}$-ring spectra $A$ su…