# Questions tagged [hochschild-cohomology]

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### 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 $k$-...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 $A''$...
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### 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 ...
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### 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 ...
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### Relation between the Hochschild cohomology of group algebras and groupoids

Is there a known relation between the Hochschild cohomology of group algebras and cohomology of groupoids? Clarification: It is known that 1-dimensional Hochschild cohomology of the Group algebra C[...
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