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Questions tagged [hochschild-cohomology]

For questions relating to the calculation or definition of Hochschild (co)homology, an algebraic invariant of associative algebras, dg algebras and dg categories.

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Replacing extension with left-right projective extensions

I'm currently reading the article "An exact sequence interpretation of the Lie bracket in Hochschild cohomology" by Stefan Schwede. The thing I don't understand is the exactness in the proof ...
Najonathan's user avatar
3 votes
2 answers
210 views

Relative Hochschild cohomology

Gerstenhaber's paper Algebraic Cohomology and Deformation Theory introduces relative Hochschild cohomology, which I have never seen before: Let $k$ be a field. Let $A$ be an associative, unital $k$-...
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Normalized Hochschild complex

For a given field $k$, let $A$ be an associative, unital $k$-algebra. Let $M$ be an $A$-bimodule. Define the Hochschild cochain complex of $A$ with coefficients in $M$ as $$C^n(A;M):=\operatorname{Hom}...
Margaret's user avatar
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1 vote
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What is a commutative infinity-algebra?

In the nLab article on Hochschild cohomology it says Thus, for $A$ a commutative $\infty$-algebra, its Hochschild homology complex is its $(\infty,1)$-tensoring $S^1\cdot A$ with the $\infty$-...
Margaret's user avatar
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An associative k-algebra whose enveloping algebra $A \otimes_k A^{op}$ is not iso to $A \otimes_k A$

I'm trying to find an example of a finite-dimensional $k$-algebra $A$ for some field $k$, ideally $\mathbb R$, such that $A \otimes_k A \not\cong A \otimes_k A^{op}$. A lot of algebras have $A \cong A^...
wlad's user avatar
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Book recommendation for Cyclic Homology

I'm starting to learn what is Cyclic homology with the Loday's book "Cyclic Homology". It is a very well book for the technical part. However, I would like also to understand the intuition ...
newuser's user avatar
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Enveloping Algebra and localization on maximal ideal.

Let $k$ be a field and $A$ a commutative $k$-algebra. The enveloping algebra of $A$, denoted by $A^e$ is defined as $A\otimes_kA^o$ where $A^o$ is the opposite algebra. Here the opposite algebra does ...
Math.mx's user avatar
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Describe the Hochschild $2$-cocycle which deforms $\mathbb{C}[x,y]$ into Weyl algebra $A_1$.

I'm attempting Exercise 5.1.15 of Witherspoon's Hochschild cohomology for Algebras. For an algebra $(A,\cdot)$, a deformation of it (over $\mathbb{C}[t]$) will be a product $\star$ on $A\otimes \...
Ted Jh's user avatar
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Seeing that a product of quasi-free algebras is quasi-free in terms of lifting homomorphisms

A $k$-algebra over a field $k$ is quasi-free if for any bimodule $M$ the second Hochschild cohomology $H^2(A,M)$ vanishes or, equivalently, if for any square-zero extension $R/M=A$ there is a lifting ...
Sergey Guminov's user avatar
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Computing the global and Hochschild dimensions of a free product of direct products of fields.

Let $k$ be a field (algebraically closed of characteristic 0, but I do not expect it to make a difference). Consider the algebra $A=k^{n+1}\ast k^{m+1}$, where $\ast$ denotes the free product of ...
Sergey Guminov's user avatar
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Well-defindness of a map involving tensor product.

Let $k$ be a field. The identity element of any $k$-algebra $\Gamma$ gives a structure map $I\colon k\to \Gamma$; its cokernel $\Gamma/I(k) = \Gamma /(k.1_{\Gamma})$ will be denoted $\overline{\Gamma}$...
User6797's user avatar
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Conceptual proof that Hochschild boundary is a derivation for the shuffle product

Let $k$ be a commutative ring with $1$ and $A$ a commutative unital $k$-algebra ($k$ and $A$ are assumed to be associative). Denote by $(C_\bullet(A),b)$ the Hochschild chain complex of $A$ and let $$ ...
Albert's user avatar
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Cohomology of algebra over an operad as naive Ext-functor instead of with Kähler module of differentials

Let $\mathcal{P}$ be an operad and $A$ a $\mathcal{P}$-algebra. In Algebraic Operads(AO) by Loday-Vallette there are some cohomology theories defined for $A$: Operadic cohomology = cohomology of $C_{...
Lilolance's user avatar
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Isomorphism of enveloping algebra homomorphisms to $k$-homomorphisms?

Given commutative ring $k$, associative $k$-algebra $A$, and $A$-bimodule $M$, we can define the enveloping algebra $A^e:=A\otimes A^{op}$, where $A^{op}$ is the algebra $A$ but with multiplication ...
Cameron's user avatar
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Interpretation of vanishing of cohomology groups

Let $G$ be a group and $M$ a left $G-$module. It is well know that for some conditions all the cohomology groups $H^{i}(G,M)$, $i=0,1,2,.....$ vanish. The same can be do for the Hochschild cohomology ...
José Luis  Camarillo Nava's user avatar
1 vote
1 answer
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Computing Hochschild cohomology groups $H^i (A,M)$ in GAP/QPA

Let $A$ be a finite-dimensional $k$-algebra. I would like to calculate the second Hochschild cohomology group $H^2(TA,DA)$, where $TA$ is the trivial extension of $A$, and $DA = \mathrm{Hom}_k(A, k)$. ...
Vasco's user avatar
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About the definition of $A^{e}-$mudule structure of an $A$-bimudule $M$

Let $R$ be a conmutative ring (with unit $1$) and $A$ an associative $R$-algebra (with unit $1_{A}$). Let $A^{e}=A\otimes_{R}A^{opp}$ the eveloping algebra of $A$. Let $M$ be an $A$-bimodule. I want ...
José Luis  Camarillo Nava's user avatar
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An example on Hochschild Cohomology Groups

I'm starting to study Hochschild cohomology from the book of Sarah Witherspoon, “Hochschild Cohomology for Algebras”. By a lecture notes of Maria Julia Redondo, “Hochschild cohomology: some methods ...
José Luis  Camarillo Nava's user avatar
1 vote
1 answer
84 views

Action of the centre on Hochschild cohomology

Let $A$ be an associative algebra, $Z(A)$ the centre of $A$, $C^n(A, A) = \mbox{Hom}_{A^e}(A^{\otimes n +2}, A)$ the $n$-th component of the Hochschild cochain complex, $b:C^n(A,A) \to C^{n+1}(A, A)$ ...
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Computing relative Hochschild (co)homology

I'm trying to compute/find simple examples the relative Hochschild cohomology of an extension of algebras. Let $$B \subseteq A$$ be an extension of algebras. Firstly there is the relative bar complex ...
Najonathan's user avatar
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1 answer
286 views

Why is Hochschild cohomology $HH^n(A, A)$ not a functor of $A$?

Let $k$ be a commutative ring, $A$ a $k$-algebra and $HH^n(A, M)$ the $n$-th Hochschild cohomology of $A$ with coefficients in the $A$-bimodule $M$. In the book Cyclic Homology by Loday the following ...
SeraPhim's user avatar
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Relation between Hochschild homology and cohomology

Let $A$ be an associative algebra, then we have the Hochschild chain complex, namely $$ \dotsb \to A^{\otimes 3} \xrightarrow{d_2} A^{\otimes 2} \xrightarrow{d_1} A \,, $$ where, for example, $d_1 (...
iou's user avatar
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3 votes
1 answer
228 views

Isomorphism of square zero extension algebras related to Hochschild cohomology

I'm trying to solve exercise 5.14 of these notes. Let $A$ be an algebra over a commutative ring $k$ and $M$ an $A$-$A$-bimodule. Define on the direct sum module $A\oplus M$ the square zero extension ...
Javi's user avatar
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11 votes
0 answers
332 views

Many definitions of Hochschild homology and cyclic homology

It appears that there are more definitions of cyclic homology than there are people working on cyclic homology. As a newcomer, this confuses me to no end. I've written a list of definitions that some ...
nagger's user avatar
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Invariance of Yoneda product

Are Yoneda products (also known as cup product) on Hochschild cohomology of two quasi-isomorphic DGAs equivalent?
Arun 's user avatar
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2 answers
403 views

Cap product for Hochschild (co)chains

Let $A$ be an associative algebra and $M$ be an $A$-bimodule. Then we can form the Hochschild cochains $C^\bullet(A,A)$ and chains $C_\bullet(A, M)$ and define a pairing (cap product) $$ C^\bullet(A,...
C_M's user avatar
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8 votes
1 answer
430 views

Computing Hochschild cohomology of an algebra in GAP

I would like to calculate Hochschild cohomology of a path algebra of a quiver (modulo some relations) using GAP. Creating the quiver and its path algebra modulo relations is explained in detail in the ...
Earthliŋ's user avatar
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3 votes
1 answer
762 views

Homology of free loop space and Hochschild cohomology

I am looking for honest proof of the following isomorphism. For a simply connected space $X$, let $LX$ be its free loop space. Then $$H_{\ast}(LX) \simeq HH^\ast(C^\ast(X),C_\ast(X))$$ I have looked ...
Allen Cho's user avatar
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1 answer
396 views

Vanishing Hochschild Cohomology over Polynomial Rings

At the moment I’m writing my master thesis and need help to understand stuff from homological algebra. Concretely I search for a reference to the following problem: Let $R = k[x_1, \dotsc, x_n]$ be a ...
qwta's user avatar
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4 votes
0 answers
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Rigidity from vanishing cohomology

Given a $k$-algebra $A$ with an associative multiplication on it $m:A\otimes A\to A$. It seems to be part of the mathematical folklore that the second Hochschild cohomology group ($HH^2(A,A)$) ...
sb1567's user avatar
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0 answers
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How to prove that the Hochschild cohomology algebra is isomorphic to the algebra given by Hom's in the derived category.

Let $A$ be a finite-dimensional algebra over an algebraically closed field (I think this can be generalized to Artinian algebras over commutative rings, but lets work with this hypothesis for now). ...
Marco Armenta's user avatar
3 votes
0 answers
80 views

Compatibility of Hochschild Cohomology

Let $A$ and $B$ be $k$-algebras with $k$ a field. Suppose that $\psi \colon A \xrightarrow{\cong} B$, and recall that $\mathrm{HH}^*(A) = \mathrm{Ext}_{A^e}^*(A,A)$ with the Gerstenhaber cup product ...
mwmjp's user avatar
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0 answers
61 views

Maps inducing identity in Hochschild and cyclic theories

Let $A$ be a unital algebra over $\mathbb{C}$, $M$ be an $A$ bimodule, $C^n(A, M)$ be a space of all $n$-linear maps $f \colon A^n \to M$ (to be called $n$-cochains) and define $b \colon C^n(A, M) \to ...
truebaran's user avatar
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5 votes
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Hochschild cohomology of a smooth manifold: a sheaf?

It can be shown that the Hochschild cohomology of the algebra of smooth function on $X$ is naturally isomorphic to the sheaf of de Rham currents. But is it possible to see from the very definition of ...
InfiniteLooper's user avatar
1 vote
0 answers
308 views

Hochschild homology of dgas with nontrivial differential

In this question, we see how to compute the Hochschild homology of a dga with zero differential: it’s just the same as computing its Hochschild homology as a graded algebra. I want to know about ...
JHF's user avatar
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5 votes
1 answer
563 views

Hochschild homology: change of ground ring

Theorem 9.1.7 in Weibel’s homological algebra reads as follows (I will change the notation slightly): Let $f \colon k \to \ell$ be a morphism of commutative rings. Denote $\otimes = \otimes_k$. Let $A$...
Bruno Stonek's user avatar
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2 votes
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Hochschild cohomology of a boolean ring

I can’t find any papers studying the Hochschild cohomology ring $H^*(B, B)$, where $B$ is a boolean ring, so I was wondering if this is known.
Mike J.'s user avatar
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4 votes
2 answers
934 views

Reference for proof of Hochschild–Kostant–Rosenberg for Hochschild cohomology

Is there a place where there is a full proof of the Hochschild–Kostant–Rosenberg theorem for Hochschild cohomology? I am aware of many places where the result is proven for Hochschild homology, i.e., ...
Anette's user avatar
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6 votes
0 answers
347 views

Deformations of associative algebras and Hochschild cohomology.

I am studying the deformation theory of associative algebras (and Poisson algebras) and came across a question for which I cannot find an answer: Let $(A, \mu)$ be a commutative associative algebra ...
unknownymous's user avatar
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5 votes
1 answer
404 views

Hochschild (co)homology and derived functors

Suppose that $A$ is (complex) unital algebra. We will consider $A-A$ bimodules $M$: such a bimodule is the same as (say) left $A \otimes A^{op}$ module. Let us define $C_n(A,M)$ as $M \otimes A^{\...
truebaran's user avatar
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8 votes
2 answers
1k views

Definition of Hochschild homology in terms of Tor functor (bar resolutions)

I had 2 kind of dumb questions about the definition of Hochschild homology in terms of the Tor functor: 1 - Let $R$ be a $k$-algebra and $M$ an $R$-bimodule, let $H_*(R,M)$ be the Hochschild homology ...
The K's user avatar
  • 355
3 votes
1 answer
174 views

Hochschild cohomology of skew polynomial rings

Definition The skew polynomial algebra over $\mathbb{C}$ is defined as $\mathbb{C}\langle x,y\rangle/(xy-yx+x)$ or alternatively as $\mathbb{C}[x,y,\sigma]$, where $\sigma$ is the automorphism on $\...
ABIM's user avatar
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4 votes
2 answers
204 views

Hochschild cohomology of a formal quantization of an associative algebra

Let $A$ be a commutative associative $k$-algebra and let $A[[\hbar]]$ be the formal deformation of $A$. I would like to know if there is a relation between the Hochschild co-homologies $\mathrm{HH}^{\...
Flavius Aetius's user avatar
5 votes
0 answers
152 views

Has this variation of Hochschild cohomology been studied?

Let $k$ be a field, and let $A$ be a commutative $k$-algebra. Let $M$ be an abelian group, and assume that it an $n$-$A$-module. That is: it has $n$ different $A$-module structures, and they are ...
user147705's user avatar
3 votes
1 answer
176 views

Why is Hochschild cohomology just a group and not a module?

This is probably a very basic question in Hochschild theory. Let $k$ be a field, and let $A$ be a $k$-algebra (which is not commutative). If $M$ is an $A$-bimodule, then the $n$-th Hochschild ...
the L's user avatar
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1 vote
0 answers
104 views

Second Hochschild cohomology and extensions

I started learning the theorem that says there is a one-to-one correspondence between $\mathrm{Ext}(A, M)$ and $H^2(A, M)$. However, the proof is not clear. I managed to show that there is a well-...
Koam Ali's user avatar
10 votes
0 answers
996 views

Morita-invariance of Hochschild (co)homology.

Ok, I’m reading the paper Homology and cohomology of associative algebras. A concise introduction to cyclic homology by Christian Kassel, and on page 19 he says that Hochschild homology is Morita-...
L-A's user avatar
  • 357
14 votes
1 answer
2k views

Hochschild homology - motivation and examples

I'm currently trying to learn about Hochschild homology of differential graded algebras. After reading the definition, the notion of Hochschild homology is somewhat unmotivated and myterious to me. ...
Dave's user avatar
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6 votes
1 answer
288 views

Wedge product of Hochschild cohomology classes in characteristic $2$

Let $A$ be a smooth commutative $k$-algebra, for $k$ a commutative ring. By the Hochschild–Kostant–Rosenberg theorem, we have that $\mathrm{HH}^*_k(A) \cong \bigwedge^* \mathrm{Der}_k(A, A)$, where $\...
KReiser's user avatar
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12 votes
1 answer
720 views

Hochschild homology of Weyl algebra

Could someone explain to me how one can compute the Hochschild homology of the Weyl algebra $A_n$ (i.e., algebra of differential operators with polynomial coefficients in $n$ variables)?
Alex's user avatar
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