# 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 ...
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### 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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### 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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### 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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### 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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### 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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