# Questions tagged [hochschild-cohomology]

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### Hochschild cohomology of an Azumaya algebra

Let $k$ be a field. Given a commutative $k$-algebra $Z$ and an associative algebra $A$ that is Azumaya over $Z$, do we have an isomorphism of Hochschild cohomologies: $HH^*(A) \cong HH^*(Z)$? This is ...
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### How do you prove that Hochschild cohomology is Morita invariant?

I am simply trying to show that $HH^\bullet(A)= HH^\bullet(M_r(A))$ for any matrix ring of $A$. In Loday's book (Sect 1.5.6) the Morita invariance is explained as follows : it says that if $M$ is an ...
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### Is there a coproduct on Hochschild homology of an algebra?

Suppose $\epsilon : A\longrightarrow K$ is an augmented algebra over a commutative ground ring $K$, so that $K$ can be treated as an $A$-bimodule. Is there a coproduct structure on the Hochschild ...
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### First and second cohomology groups of Banach algebras

Johnson in the introduction section (page 1) in "Cohomology in Banach algebras" ZBL0256.18014, wrote that Guichardet in [14,15] obtained for a Banach algebra $A$, one has $H^1(A,X)=H^2(A,X)=0$, ...
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### Question on vanishing Hochschild cohomology

Recall that for an $K$-algebra $A$ with $A^e:=A^{op} \otimes_K A$ the Hochschild cohomology is defined as $HH^n(A,M):=Ext_{A^e}^n(A,M)$. Question: Is there a finite dimensional selfinjective ...
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### A 2- cocycle $\tau$ which is not cyclic but it still satisfies the stability of $\tau(e,e,e)$ for idempotent $e$

I learned the following statement from page $20$ of the book Noncommutative Geometry by Alain Connes: Let $\tau$ be a $2$-cyclic cocyle on a $C^*$ algebra. Then for every smooth curve $e(t)$ of ...
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### Units in the (stable) center of a Frobenius algebra [duplicate]

Let $A$ be a Frobenius algebra with center $Z(A)$ and $I\subset Z(A)$ the ideal of elements in the image of some $A$-bimodule map $A\rightarrow A\otimes A\rightarrow A$, where the second map is ...
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### Why is every deformation of the universal enveloping algebra of a complex semisimple Lie algebra trivial?

I have read in these lecture notes that every deformation $U_h(\mathfrak{g})$ of $U(\mathfrak{g})$ is trivial, i.e. isomorphic to $U(\mathfrak{g})[[h]]$ as associative $\mathbb{C}[[h]]$-algebras. Why ...
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### Higher traces in Hochschild cohomology

Let $A$ be an associative algebra over a field $k$. Let $\rho:A \rightarrow \mathrm{End}(M)$ a left module, finite dimensional over $k$. Then the map $a \mapsto \mathrm{tr}_M \rho(a)$ is a well ...
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### intuition for hochschild homology

According to this post Intuition for group homology, I wonder what is the intuition for Hochschild homology. The Hochschild homology is defined as the homology of this complex chain. Given a ...
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### A differential graded Lie algebra with the Hochschild differential

Let $(V,\cdot)$ be an associative algebra and $W$ be a vector space endowed with a bimodule structure $\triangleright:V\otimes W\to W$ and $\triangleleft:W\otimes V\to W$ such that the following ...
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### Morita equivalence and isomorphisms in cohomology theories

Let $A,B$ be two unital algebras. We say that $A,B$ are Morita equivalent if there are $A-B$ and $B-A$ bimodules $P,Q$ such that $$P \otimes_{B} Q \cong A, Q \otimes_A P \cong B$$ (as $A-A$ and $B-B$ ...
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### First Hochschild cohomology of $A=K[x]/(x^n)$

Given the algebra $A=K[x]/(x^n)$ for some field $K$ and natural number $n \geq 2$ with enveloping algebra $A^e=A \otimes_K A$. It is easy to see that the 1. Hochschild cohomology of $A$ is nonzero ...
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### Hochschild cohomology of certain local algebras

Let $K$ be a field and $A$ the algebra $K\langle x_1,...,x_n\rangle/J^m$ for $n \geq 1$ and $m \geq 2$, where $K\langle x_1,...,x_n\rangle$ is the non-commutative polynomial ring in $n$ variables over ...
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### Lie algebra of a p-group

Given a p-group P, the first hochschild cohomology of the group algebra (over a field of characteristic p) of P is a nonzero Lie algebra. Is it known what Lie algebra results depending on P? I have no ...
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### Another question on Hochschild cohomology

All algebras are assumed to be connected over a field. Given an algebra $A$ with minimal faithful projective-injective left module $Af$ for some idempotent f. Let $M$ be an $A$-module of infinite ...
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### A question on Hochschild cohomology

Given a nonsemisimple symmetric algebra B and a non-selfinjective algebra A (all algebras are finite dimensional over a field and connected). Can A and B have isomorphic Hochschild-cohomology rings?
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### Hochschild coboundary on the space of alternative forms

Assume that $A$ is a complex algebra. By $C^{n}(A)$ we mean the space of all $n-$linear map $\phi:A^n \to \mathbb{C}$. An alternative $k-$ form is an element $\phi \in C^{k}(A)$ ...
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### Normalization of Hochschild cocycles

Let $A$ be a unital algebra over $\mathbb{C}$. Let $C^n(A)$ be the space of all $n+1$-linear maps $f:A^{n+1} \to \mathbb{C}$ (to be called $n$-cochains). Define $b:C^n(A) \to C^{n+1}(A)$ by the ...
Let $A$ be a unital algebra and $u \in A$ be an invertible element. Let us consider $u_n(a_0 \otimes a_1 \otimes ... \otimes a_n):=ua_0u^{-1} \otimes ua_1u^{-1} \otimes ... \otimes ua_nu^{-1}$. Then $(... 1answer 536 views ### Isomorphism in cyclic cohomology vs isomorphism in Hochschild cohomology Let$A$be a unital algebra over a field$K$,$C^n(A)$a space of all$n+1$linear maps into scalar field$k$(I'm interested in case$k=\mathbb{C}$) and $$(bf)(a_0,...,a_{n+1})=\sum_{i=0}^n(-1)^if(... 1answer 392 views ### Can the Hochschild cochain complex be given the structure of a “homotopy BV algebra”? In a 1993 letter, Deligne posed the following (paraphrased from a paper of Gerstenhaber and Voronov's): Conjecture (Deligne). The Hochschild cochain complex CC^*(A) of an associative algebra ... 0answers 235 views ### Hochschild cohomology of SU(2) I have a question about the computation of an Hochschild Cohomology. Or at least about a space which really looks like a cohomology space. All the functions i consider are assumed to be smooth. Let's ... 2answers 433 views ### Gerstenhaber conjecture for free loop space I- Is the following statement still a conjecture see this article ? Conjecture (?) Let M be a simply connected compact oriented d-manifold (smooth), then HH^{\ast}(C^{\ast}(M)) the Hochschild ... 0answers 883 views ### What is the Hochschild cohomology of the Fukaya-Seidel category? Let (Y, \omega) be a compact symplectic manifold and let Fuk(X,\omega) be its Fukaya category. The Hochschild cohomology of this category should be given by HH^\bullet(Fuk(Y,\omega))=H^\bullet(Y, ... 0answers 268 views ### Hochschild Cohomology of the Quantum Torus I would like some advice on how to compute directly, or by a higher powered method the Hochschild Cohomology groups of the quantum torus using the stated complex I have found. I think there are ... 1answer 147 views ### Antisymmetrization of the Hochschild cocycle Let A be a commutative (unital, complex) algebra and let \varphi be a n+1-linear functional on A (we will call it cochain). Define$$(b\varphi)(a_0,a_1,...,a_{n+1}):=\sum_{j=0}^{n}(-1)^j\... 0answers 393 views ### Contraction with a vector field and pullback bundle While trying to understand the proof of "smooth" version of Kostant-Hochschild-Rosenberg theorem (which is due to Connes for compact smooth manifolds) I found the following argumentation: one is ... 0answers 171 views ### Hochschild cohains-type models for smooth functions on shifted cotangent bundles Let$M$be a smooth manifold. Then, by the well known Hochschild-Kostant-Rosenberg theorem, the cochain complex$C^*(C^\infty(M))$of Hochshild cochains on the algebra$C^\infty(M)$of smooth ... 2answers 1k views ### Relationship between Hochschild cohomology and Drinfeld centers Let$HH_*(A,N)$(or$HH^*(A,N)$) be the Hochschild homology (or cohomology) of an associative algebra$A$with coefficients in an$A$-bimodule$N$. I was reading nlab's entry on Hochschild cohomology ... 1answer 195 views ### Hochschild cohomology of the skew group ring D(X)#G in the complex analytic case Let$X$be a n-dimensional complex (not necessarily compact) manifold, let$G$be a finite subgroup of$Aut(X)$acting by biholomorphic maps on$X$and let$D(X)$be the ring of differential operators ... 1answer 227 views ### Hochschild cohomology of commutative quotients Notation: Let$k$be a commutative local ring and let$HH^{i}(A,N)$denote the$i^{th}$Hochschild cohomology$k$-module of a$k$-algebra A with coefficients in an$(A,A)$-bi-module$N$. If$x:=\{...
In all the literature I've come across there are many concrete interpretations of the first few Hochschild Cohomology groups. For example $HH^1(A,M)\cong Derivation/Inner Derivations$ etc.... In ...