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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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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Kronecker pairing for Hochschild (co)homology

Let $A$ be an unital associative algebra over a commutative ring $k$ and denote by $A^{e}$ its enveloping algebra. Let $M$ and $M'$ be two $A$-bimodules that are symmetric as $k$-bimodules. Denote by $...
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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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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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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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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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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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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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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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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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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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