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.
49
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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^...
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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 ...
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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 ...
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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 \...
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Hochschild homology of stable categories as topological chiral homology
Let $\mathscr{C}_0$ be a small idempotent complete stable category tensored over some symmetric monoidal category $\mathcal{E}$.
Its Ind-completion $\mathscr{C} := \operatorname{Ind}(\mathscr{C}_0)$ ...
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Hochschild cohomology and de Rham cohomology
I know that given a real finite dimensional associative algebra $A$, its automorphism group $\text{Aut}(A)$ is a Lie group. Is there some relation between the de Rham cohomology of this Lie group and ...
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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 ...
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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 ...
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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 $\overline{\Gamma}$...
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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
$$
...
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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_{...
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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 ...
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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 ...
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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)$.
...
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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 ...
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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 ...
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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$$ be an extension of algebras. Firstly there is the relative bar complex ...
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Why is Hochschild cohomology $HH^n(A, A)$ not a functor of $A$?
Let $k$ be a commutative ring, $A$ a $k$-algebra and $HH^n(A, M)$ the $n$-th Hochschild cohomology of $A$ with coefficients in the $A$-bimodule $M$. In the book Cyclic Homology by Loday the following ...
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Relation between Hochschild homology and cohomology
Let $A$ be an associative algebra, then we have the Hochschild chain complex, namely
$$
\dotsb \to A^{\otimes 3} \xrightarrow{d_2} A^{\otimes 2} \xrightarrow{d_1} A \,,
$$
where, for example, $d_1 (...
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Isomorphism of square zero extension algebras related to Hochschild cohomology
I'm trying to solve exercise 5.14 of these notes.
Let $A$ be an algebra over a commutative ring $k$ and $M$ an $A$-$A$-bimodule. Define on the direct sum module $A\oplus M$ the square zero extension ...
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Many definitions of Hochschild homology and cyclic homology
It appears that there are more definitions of cyclic homology than there are people working on cyclic homology. As a newcomer, this confuses me to no end. I've written a list of definitions that some ...
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Invariance of Yoneda product
Are Yoneda products (also known as cup product)
on Hochschild cohomology of two quasi-isomorphic DGAs equivalent?
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Cap product for Hochschild (co)chains
Let $A$ be an associative algebra and $M$ be an $A$-bimodule. Then we can form the Hochschild cochains $C^\bullet(A,A)$ and chains $C_\bullet(A, M)$ and define a pairing (cap product)
$$
C^\bullet(A,...
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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 ...
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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 ...
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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 ...
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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 ($HH^2(A,A)$) ...
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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).
...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $A$...
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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.
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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., ...
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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 ...
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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^{\...
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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 ...
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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 $\...
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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}^{\...
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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 ...
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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 ...
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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-...
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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-...
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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. ...
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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 $\...
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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)?
- 6,090
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What is the connection between Grothendieck's Differential Operators and Hochschild Cohomology
For a given commutative algebra $A$ over a field $\mathbb{K}$(with char=0) the algebra of differential operators on $A$ is the set of endomorphism $D$ of $A$ such for some $n$ we have that for any ...
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