Skip to main content

Questions tagged [hochschild-cohomology]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1 vote
0 answers
87 views

Cyclic homology with coefficients in a bimodule

I've recently been trying to understand Hochschild and cyclic co/homology better, and I've noticed that while it's common to define the Hochschild homology $\mathrm{HH}_{\bullet}(A;M)$ of an $R$-...
Emily's user avatar
  • 11.4k
3 votes
1 answer
145 views

Minimality of the Koszul resolution

Let $R = \mathbb{C}[x,y]$ and $V = \mathbb{C}x\oplus\mathbb{C}y$. Then, the Koszul resolution of $R$ (as an $R$-bimodule) is given by \begin{align*} 0\to R\otimes_{\mathbb{C}}\wedge^2V\otimes_{\mathbb{...
Qwert Otto's user avatar
3 votes
0 answers
78 views

Drinfeld centres and formal moduli problems

If $\mathcal{P}$ is a sufficiently nice operad, then by [Higher Algebra, 5.3] you can form its centre: $$\mathcal{Z}_{\mathcal{P}}\ :\ \mathcal{P}\text{-Alg}\ \to\ \mathbf{E}_1\text{-Alg}(\mathcal{P}\...
Pulcinella's user avatar
  • 5,565
6 votes
2 answers
467 views

Is the exterior algebra intrinsically formal?

Following 4.6 and 4.7 of this paper by Seidel and Thomas, a graded algebra $A$ is called intrinsically formal if any two dgas with cohomology $A$ are quasi-isomorphic. There is a sufficient condition ...
Faniel's user avatar
  • 653
9 votes
1 answer
384 views

Hochschild cohomology of a group algebra

Let $K$ be a field and $G=\pi_1(\Sigma_g)$ the surface group of genus $\geq 2$. I want to know the Hochschild cohomology of the group algebra $A=K[G]$ with coefficients in $A$ and $A\otimes A$, namely,...
Qwert Otto's user avatar
7 votes
0 answers
133 views

On the invariance of the Kaledin class

In Formality of DG algebras (after Kaledin), Lunts introduces an $A_\infty$-Hochschild cohomology class, called the Kaledin class, controlling formality of an $A_\infty$-algebra up to a certain order. ...
domenico fiorenza's user avatar
4 votes
0 answers
135 views

Hochschild cohomology of path algebra as a cohomology of simplicial complex

M. Gerstenhaber and S. D. Schack have shown that a cohomology of simplicial complex can be expressed as a Hochschild cohomology of path algebra constructed from this complex (link). Is the opposite ...
Alexander's user avatar
1 vote
0 answers
80 views

Gerstenhaber bracket for Hochschild cohomology with values in a module

I am currently trying to compute obstructions in a Hochschild cohomology $\mathrm{HH}^* (A,M)$ where $A$ is a $\Bbbk$-algebra and $M$ an $A$-bimodule. The obstruction I am looking at looks a lot like ...
Felix's user avatar
  • 203
8 votes
2 answers
674 views

How to compute the periodic cyclic homology of this algebra

Let $k=\mathbb{C}$ be the field of complex numbers. I consider the (DG) algebra $A:=k[x]/(x^2)$ such that $\deg(x)=-1$. My question is how to compute the periodic cyclic homology, Hochschild homology ...
user41650's user avatar
  • 1,952
4 votes
0 answers
128 views

Transferred $L_\infty$-structure from Hochschild dgLA

Let $D_{poly}$ be the differential graded Lie algebra (dgLA) of differentiable Hochschild cochains on a manifold $\mathscr M$, endowed with the usual Gerstenhaber bracket $[-,-]_G$ and Hochschild ...
thingsthatmighthavebeen's user avatar
5 votes
2 answers
309 views

Cubical vs. simplicial Hochschild cohomology

Simplicial Hochschild cohomology. $\newcommand{\Hom}{\mathrm{Hom}}\newcommand{\B}{\mathrm{B}}\newcommand{\Obj}{\mathrm{Obj}}\newcommand{\HH}{\mathrm{HH}}\newcommand{\Mod}{\mathsf{Mod}}$One way to ...
Emily's user avatar
  • 11.4k
3 votes
1 answer
368 views

Is there an analogue of the module of differentials for "higher order derivations" in the Hochschild/cyclic senses?

Note added by YC: the definition below of the cyclic sub-complex is incorrect; and the "higher order derivations" referred to here are traditionally known (since the 1940s) as n-cocycles. $\...
Emily's user avatar
  • 11.4k
5 votes
0 answers
108 views

Hochschild cohomology of reductive algebraic groups

I have two questions about the Hochschild cohomology of algebraic groups. The first one will reveal the depth of my ignorance, and, because of this depth, I might be asking a question more ...
LSpice's user avatar
  • 11.7k
7 votes
1 answer
598 views

Relation between symplectic (co)homology and Hochschild (co)homology and deformations

A very fluffy question in which I'm ignorant of homology/cohomology, grading etc: The open-closed and closed-open string maps relating the symplectic (co)homology and Hochschild (co)homology of the ...
86846515312's user avatar
3 votes
0 answers
152 views

Hochschild cohomology and outer automorphisms

Given an algebra $A$, I believe that the "conjugation" action of $\mathrm{Aut}(A)$ on $\mathrm{HH}^*(A)$ factors through $\mathrm{Out}(A)$. I’m looking for a reference.
Fernando Muro's user avatar
1 vote
0 answers
144 views

Two definitions for smoothness

I'm currently reading Sarah Witherspoon's book on Hochschild Cohomology. At the beginning of the fourth chapter it is given the following definition: Definition 1. If $k$ is a field and $A$ is a $k$-...
cos_dm_math21's user avatar
2 votes
1 answer
383 views

Can we compute the Hochschild cohomology for $k[x]$ through the Hochschild complex?

For an algebra $A$ we can define its Hochschild cohomology $HH^{\bullet}(A,A)$ as in this wikipedia page. Now let $A=k[x]$ be the polynomial ring where $k$ is a field. It is well-known that $HH^{0}(A,...
Zhaoting Wei's user avatar
  • 8,737
1 vote
0 answers
248 views

Does the Hochschild cohomology of an $A_{\infty}$-algebra have an algebra structure?

For an algebra $A$ we can define its Hochschild cohomology (see this Wikipedia page) $HH^{\cdot}(A,A)$. It is well-known that the cup product makes $HH^{\cdot}(A,A)$ a (graded-commutative) algebra. ...
Zhaoting Wei's user avatar
  • 8,737
6 votes
1 answer
397 views

Hochschild cohomology of group ring of a free group

Let $G$ be a free group of finite rank. Consider any commutative ring $R$ containing $\mathbb{Z}$. Consider the group ring $RG$. Q) What can we say about the Hochschild cohomology groups of $RG$ with ...
Cusp's user avatar
  • 1,703
2 votes
0 answers
206 views

Hochschild cohomology of a sheaf of associative algebras

Assume that $X$ is a complex manifold. Let $\delta: X\to X\times X$ be the diagonal map. Assume that $\mathcal{A}_X$ is a $\mathbb C_X$-algebra and $\mathcal{M}_X$ is a left $\mathcal{A}_X\otimes_{\...
Flavius Aetius's user avatar
3 votes
1 answer
199 views

Hochschild cohomology of finite semisimple algebras

Let $A$ be a finite semisimple algebra over $\mathbb{k}$, a perfect field. Is true that the second Hochschild cohomology group vanishes, i.e. $$HH^2(A) = 0?$$ In order to make this question a little ...
JeCl's user avatar
  • 1,001
4 votes
0 answers
105 views

Explicit $L_\infty$-operations on Hochschild cochains of $A_\infty$-algebra

It is well-known that the Hochschild cochain complex $\mathrm{CC}^*(A)$ of an associative algebra $A$ carries a lot of structure. In particular: a differential, a cup product, and a bracket, which ...
Jack Smith's user avatar
3 votes
0 answers
255 views

On the Hochschild cohomology of the minimal model of an $A_\infty$ algebra

Suppose $(A, (\mu_k))$ is a (curved) $A_\infty$ algebra, and let $(\tilde A, (\tilde\mu_k))$ be its minimal model. Now, we have two Hochschild cohomology rings $HH^*(A)$ and $HH^*(\tilde A)$. (It may ...
Hang's user avatar
  • 2,729
0 votes
0 answers
324 views

Proof of “Hochschild-Serre” spectral sequence

I'm looking for a detailed proof of the Hochschild-Serre spectral sequence for Galois Cohomology: If we have $H$ a normal subgroup of $G$ (profinite group) and $T$ is a $G-$module, we get: $$ 0\...
marco's user avatar
  • 109
8 votes
0 answers
286 views

Which algebra is $\mathbb{F}_p \otimes_{HH^{\cdot} (\mathbb{F}_p)} \mathbb{F}_p$?

Let $HH^{\cdot}(\mathbb{F}_p)$ be the Hochschild cohomology of $\mathbb{F}_p$ over $\mathbb{Z}$, which as a $E_1$ ring is simply $\mathbb{F}_p[x]$ with $x$ in cohomological degree $2$. Then it's clear ...
davik's user avatar
  • 2,035
6 votes
1 answer
366 views

What is the topological Hochschild cohomology of $\mathbb{F}_p$?

Following the computation of the THH (topological Hochschild homology) of $\mathbb{F}_p$ as outlined in Krause-Nikolaus. We use the fact that $\mathbb{F}_p$ is initial $E_2$ ring with $0=p$ to compute ...
davik's user avatar
  • 2,035
4 votes
0 answers
194 views

divided powers of a deformation class

Let $A$ be a (unital, associative) $k$-algebra where $k$ is a field. Given a flat deformation of $A$ one gets the deformation class $h$ in the second Hochschild cohomology $HH^2(A)$. Suppose $k$ has ...
Roman's user avatar
  • 1,526
1 vote
0 answers
68 views

Bound on Hochschild dimension of a dg-algebra

Consider a dg-algebra $A$, is there any way I can estimate the Hochschild dimension, or global dimension of $A$? More precisely the algebra that I am considering is the Endomorphism dg-algebra $\...
Felix's user avatar
  • 203
9 votes
0 answers
357 views

A characterisation of symmetric algebras using Hochschild (co)homology

A finite dimensional (connected if needed) $K$-algebra $A$ over a field $K$ is called symmetric when $A \cong Hom_K(A,K)$ as $A$-bimodules. Symmetric algebras are Frobenius algebras and include for ...
Mare's user avatar
  • 26.3k
5 votes
2 answers
356 views

Checking if Hochschild cohomology $\mathit{HH}^2(A)=0$

I am trying to compute the Hochschild cohomology of a particular bound quiver path algebra. The quiver $Q$ consists of one vertex and four loops $x,y, h_1,h_2$, and the relations $I$ are generated by: ...
Sergey Guminov's user avatar
3 votes
0 answers
83 views

Degeneration of spectral sequence computing Hochschild cohomology of enveloping algebra of Lie algebroid

Let $L$ be a Lie algebroid on a smooth affine $k$-scheme $X=spec(R)$. Recall that by definition $L$ is a locally free sheaf with the structure of a sheaf of $k$ Lie algebras, so that there exists a ...
user avatar
8 votes
0 answers
531 views

Why is Hochschild homology interesting if its cohomology groups are infinite-dimensional?

I am trying to understand Hochschild homology, in particular the Hochschild–Kostant–Rosenberg theorem. As far as I understand this result gives an isomorphism between the algebraic (Kähler) ...
Jake Wetlock's user avatar
  • 1,154
5 votes
1 answer
246 views

Hochschild cohomology of an Azumaya algebra

Let $k$ be a field. Given a commutative $k$-algebra $Z$ and an associative algebra $A$ that is Azumaya over $Z$, do we have an isomorphism of Hochschild cohomologies: $HH^*(A) \cong HH^*(Z)$? This is ...
MathManiac's user avatar
9 votes
2 answers
733 views

B-model and Hochschild cohomology

In "On the Classification of Topological Field Theories" in Example 1.4.1, Lurie introduces the B-model with target an (even dimensional) Calabi-Yau variety $X$: The Hochschild cohomology $\...
Markus Zetto's user avatar
3 votes
0 answers
125 views

Applying a Hochschild cocycle to a Maurer-Cartan element: how one should think of this?

Let $C^{\bullet}(A,M)$ be the Hochschild cochain complex of a DG-algebra $A$ with coefficients in a DG-bimodule $M$. Let $\zeta \in C^0(A,M)$ be a cocycle. Let $a \in A$ be a Maurer-Cartan element, $d(...
Dasha Poliakova's user avatar
7 votes
2 answers
812 views

How do you prove that Hochschild cohomology is Morita invariant?

I am simply trying to show that $HH^\bullet(A)= HH^\bullet(M_r(A))$ for any matrix ring of $A$. In Loday's book (Sect 1.5.6) the Morita invariance is explained as follows : it says that if $M$ is an ...
HochsMorita's user avatar
4 votes
1 answer
383 views

First and second cohomology groups of Banach algebras

Johnson in the introduction section (page 1) in "Cohomology in Banach algebras" ZBL0256.18014, wrote that Guichardet in [14,15] obtained for a Banach algebra $A$, one has $H^1(A,X)=H^2(A,X)=0$, ...
Albert harold's user avatar
2 votes
0 answers
82 views

Gerstanharber bracket and derived Hom

Let $A$ be a honest algebra or more generally, a DG algebra. It is known that the Hochschild cochain complex is quasi-isomorphic to the derived Hom complex, i.e. one has $$\mathrm{HH}^{\bullet}(A,\,A)...
Yining Zhang's user avatar
6 votes
0 answers
146 views

Hochschild cohomology of the $A_\infty$-category of paths

I would like to describe the Hochschild cohomology (in the sense of $A_\infty$-categories) of the following $A_\infty$-category associated to a topological space $X$: It has points of $X$ as objects. ...
user11267981's user avatar
3 votes
2 answers
185 views

Two definitions of minimal models

Is there any relationship between both definitions of minimal models? (the couple of definitions I know are the one mentioned in Lefèvre's thesis, in the sense that the differential is zero, and the ...
Victor TC's user avatar
  • 795
4 votes
0 answers
146 views

A relation between Hochschild cohomology of a $C^*$ algebra and its bidual

Let $A$ be a $C^*$ algebra and $A''$ be its bidual with the Arens product. Is there any relation between the Hochschild cohomolgy of $A$ with complex coefficients and the Hochschild cohomology of $A''$...
Ali Taghavi's user avatar
0 votes
1 answer
91 views

Compute the cohomology of $\mathrm{Hom} (\Omega^*(M),\Omega^*(M))$

Let $M$ be a compact smooth manifold. And particularly I am interested in the case the torus $M=T^n$. Consider the de Rham complex $(\Omega^*(M), d)$ and the cochain complex $$ C:=\mathrm{Hom} (\...
Hang's user avatar
  • 2,729
1 vote
0 answers
136 views

Question on vanishing Hochschild cohomology

Recall that for an $K$-algebra $A$ with $A^e:=A^{op} \otimes_K A$ the Hochschild cohomology is defined as $HH^n(A,M):=Ext_{A^e}^n(A,M)$. Question: Is there a finite dimensional selfinjective ...
Mare's user avatar
  • 26.3k
1 vote
0 answers
61 views

A 2- cocycle $\tau$ which is not cyclic but it still satisfies the stability of $\tau(e,e,e)$ for idempotent $e$

I learned the following statement from page $20$ of the book Noncommutative Geometry by Alain Connes: Let $\tau$ be a $2$-cyclic cocyle on a $C^*$ algebra. Then for every smooth curve $e(t)$ of ...
Ali Taghavi's user avatar
2 votes
0 answers
74 views

Units in the (stable) center of a Frobenius algebra [duplicate]

Let $A$ be a Frobenius algebra with center $Z(A)$ and $I\subset Z(A)$ the ideal of elements in the image of some $A$-bimodule map $A\rightarrow A\otimes A\rightarrow A$, where the second map is ...
Fernando Muro's user avatar
13 votes
2 answers
672 views

Why is every deformation of the universal enveloping algebra of a complex semisimple Lie algebra trivial?

I have read in these lecture notes that every deformation $U_h(\mathfrak{g})$ of $U(\mathfrak{g})$ is trivial, i.e. isomorphic to $U(\mathfrak{g})[[h]]$ as associative $\mathbb{C}[[h]]$-algebras. Why ...
cantwellnc's user avatar
11 votes
0 answers
889 views

Higher traces in Hochschild cohomology

Let $A$ be an associative algebra over a field $k$. Let $\rho:A \rightarrow \mathrm{End}(M)$ a left module, finite dimensional over $k$. Then the map $a \mapsto \mathrm{tr}_M \rho(a)$ is a well ...
Reimundo Heluani's user avatar
1 vote
0 answers
170 views

The comparison of certain modules arising from the Cauchy-Riemann differential operator

Let $\Gamma=C^{\infty}(\mathbb{R}^2)$ be the space of all smooth complex valued functions on the plane. We define the following Cauchy Riemann differential operator $D$ on $\Gamma$: $$D:\Gamma \...
Ali Taghavi's user avatar
4 votes
1 answer
259 views

A precise definition of contractible Banach algebras

I asked this question at MSE but I did not received any answer. So I ask it here at MO I am sorry if this question is elementary: What is a precise definition of a contractible Banach ...
Ali Taghavi's user avatar
3 votes
0 answers
75 views

Notion of "strict $A_\infty$ centre"

There is definition of "$A_\infty$ Centre" in article The A_\infty-Centre of the Yoneda Algebra and the Characteristic Action of Hochschild Cohomology on the Derived Category at p.28. It can be ...
Mykola Pochekai's user avatar