Questions tagged [hochschild-cohomology]
The hochschild-cohomology tag has no usage guidance.
101
questions
1
vote
0
answers
114
views
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 $k$-...
- 173
2
votes
1
answer
148
views
Can we compute the Hochschild cohomology for $k[x]$ through the Hochschild complex?
For an algebra $A$ we can define its Hochschild cohomology $HH^{\bullet}(A,A)$ as in this wikipedia page.
Now let $A=k[x]$ be the polynomial ring where $k$ is a field. It is well-known that $HH^{0}(A,...
- 8,117
1
vote
0
answers
151
views
Does the Hochschild cohomology of an $A_{\infty}$-algebra have an algebra structure?
For an algebra $A$ we can define its Hochschild cohomology (see this Wikipedia page) $HH^{\cdot}(A,A)$. It is well-known that the cup product makes $HH^{\cdot}(A,A)$ a (graded-commutative) algebra.
...
- 8,117
6
votes
1
answer
243
views
Hochschild cohomology of group ring of a free group
Let $G$ be a free group of finite rank. Consider any commutative ring $R$ containing $\mathbb{Z}$. Consider the group ring $RG$.
Q) What can we say about the Hochschild cohomology groups of $RG$ with ...
- 1,653
1
vote
0
answers
90
views
Hochschild cohomology of a sheaf of associative algebras
Assume that $X$ is a complex manifold. Let $\delta: X\to X\times X$ be the diagonal map. Assume that $\mathcal{A}_X$ is a $\mathbb C_X$-algebra and $\mathcal{M}_X$ is a left $\mathcal{A}_X\otimes_{\...
- 493
3
votes
1
answer
118
views
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 ...
- 741
4
votes
0
answers
79
views
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 ...
- 141
3
votes
0
answers
221
views
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 ...
- 2,611
0
votes
0
answers
197
views
Proof of “Hochschild-Serre” spectral sequence
I'm looking for a detailed proof of the Hochschild-Serre spectral sequence for Galois Cohomology:
If we have $H$ a normal subgroup of $G$ (profinite group) and $T$ is a $G-$module, we get:
$$
0\...
- 9
8
votes
0
answers
225
views
Which algebra is $\mathbb{F}_p \otimes_{HH^{\cdot} (\mathbb{F}_p)} \mathbb{F}_p$?
Let $HH^{\cdot}(\mathbb{F}_p)$ be the Hochschild cohomology of $\mathbb{F}_p$ over $\mathbb{Z}$, which as a $E_1$ ring is simply $\mathbb{F}_p[x]$ with $x$ in cohomological degree $2$. Then it's clear ...
- 1,723
6
votes
1
answer
311
views
What is the topological Hochschild cohomology of $\mathbb{F}_p$?
Following the computation of the THH (topological Hochschild homology) of $\mathbb{F}_p$ as outlined in Krause-Nikolaus.
We use the fact that $\mathbb{F}_p$ is initial $E_2$ ring with $0=p$ to compute
...
- 1,723
4
votes
0
answers
156
views
divided powers of a deformation class
Let $A$ be a (unital, associative) $k$-algebra where $k$ is a field. Given a flat deformation of $A$ one gets the deformation class $h$ in the second Hochschild cohomology $HH^2(A)$. Suppose $k$ has ...
- 1,253
1
vote
0
answers
57
views
Bound on Hochschild dimension of a dg-algebra
Consider a dg-algebra $A$, is there any way I can estimate the Hochschild dimension, or global dimension of $A$?
More precisely the algebra that I am considering is the Endomorphism dg-algebra $\...
- 193
9
votes
0
answers
312
views
A characterisation of symmetric algebras using Hochschild (co)homology
A finite dimensional (connected if needed) $K$-algebra $A$ over a field $K$ is called symmetric when $A \cong Hom_K(A,K)$ as $A$-bimodules. Symmetric algebras are Frobenius algebras and include for ...
- 22.1k
5
votes
2
answers
289
views
Checking if Hochschild cohomology $\mathit{HH}^2(A)=0$
I am trying to compute the Hochschild cohomology of a particular bound quiver path algebra. The quiver $Q$ consists of one vertex and four loops $x,y, h_1,h_2$, and the relations $I$ are generated by:
...
- 573
3
votes
0
answers
61
views
Degeneration of spectral sequence computing Hochschild cohomology of enveloping algebra of Lie algebroid
Let $L$ be a Lie algebroid on a smooth affine $k$-scheme $X=spec(R)$. Recall that by definition $L$ is a locally free sheaf with the structure of a sheaf of $k$ Lie algebras, so that there exists a ...
- 1,675
8
votes
0
answers
348
views
Why is Hochschild homology interesting if its cohomology groups are infinite-dimensional?
I am trying to understand Hochschild homology, in particular the Hochschild–Kostant–Rosenberg theorem. As far as I understand this result gives an isomorphism between the algebraic (Kähler) ...
- 567
5
votes
0
answers
116
views
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 ...
- 370
9
votes
2
answers
597
views
B-model and Hochschild cohomology
In "On the Classification of Topological Field Theories" in Example 1.4.1, Lurie introduces the B-model with target an (even dimensional) Calabi-Yau variety $X$: The Hochschild cohomology $\...
- 601
3
votes
0
answers
112
views
Applying a Hochschild cocycle to a Maurer-Cartan element: how one should think of this?
Let $C^{\bullet}(A,M)$ be the Hochschild cochain complex of a DG-algebra $A$ with coefficients in a DG-bimodule $M$. Let $\zeta \in C^0(A,M)$ be a cocycle. Let $a \in A$ be a Maurer-Cartan element, $d(...
- 735
7
votes
2
answers
534
views
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 ...
- 71
4
votes
1
answer
350
views
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$, ...
- 565
2
votes
0
answers
70
views
Gerstanharber bracket and derived Hom
Let $A$ be a honest algebra or more generally, a DG algebra. It is known that the Hochschild cochain complex is quasi-isomorphic to the derived Hom complex, i.e. one has
$$\mathrm{HH}^{\bullet}(A,\,A)...
- 631
6
votes
0
answers
122
views
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.
...
- 101
3
votes
2
answers
164
views
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 ...
- 795
4
votes
0
answers
138
views
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 $A''$...
- 207
0
votes
1
answer
81
views
Compute the cohomology of $\mathrm{Hom} (\Omega^*(M),\Omega^*(M))$
Let $M$ be a compact smooth manifold. And particularly I am interested in the case the torus $M=T^n$.
Consider the de Rham complex $(\Omega^*(M), d)$ and the cochain complex
$$
C:=\mathrm{Hom} (\...
- 2,611
1
vote
0
answers
124
views
Question on vanishing Hochschild cohomology
Recall that for an $K$-algebra $A$ with $A^e:=A^{op} \otimes_K A$ the Hochschild cohomology is defined as $HH^n(A,M):=Ext_{A^e}^n(A,M)$.
Question:
Is there a finite dimensional selfinjective ...
- 22.1k
1
vote
0
answers
56
views
A 2- cocycle $\tau$ which is not cyclic but it still satisfies the stability of $\tau(e,e,e)$ for idempotent $e$
I learned the following statement from page $20$ of the book Noncommutative Geometry by Alain Connes:
Let $\tau$ be a $2$-cyclic cocyle on a $C^*$ algebra. Then for every smooth curve $e(t)$ of ...
- 207
2
votes
0
answers
69
views
Units in the (stable) center of a Frobenius algebra [duplicate]
Let $A$ be a Frobenius algebra with center $Z(A)$ and $I\subset Z(A)$ the ideal of elements in the image of some $A$-bimodule map $A\rightarrow A\otimes A\rightarrow A$, where the second map is ...
- 14.1k
13
votes
2
answers
514
views
Why is every deformation of the universal enveloping algebra of a complex semisimple Lie algebra trivial?
I have read in these lecture notes that every deformation $U_h(\mathfrak{g})$ of $U(\mathfrak{g})$ is trivial, i.e. isomorphic to $U(\mathfrak{g})[[h]]$ as associative $\mathbb{C}[[h]]$-algebras. Why ...
- 175
11
votes
0
answers
755
views
Higher traces in Hochschild cohomology
Let $A$ be an associative algebra over a field $k$. Let $\rho:A \rightarrow \mathrm{End}(M)$ a left module, finite dimensional over $k$. Then the map $a \mapsto \mathrm{tr}_M \rho(a)$ is a well ...
- 3,072
1
vote
0
answers
162
views
The comparison of certain modules arising from the Cauchy-Riemann differential operator
Let $\Gamma=C^{\infty}(\mathbb{R}^2)$ be the space of all smooth complex valued functions on the plane. We define the following Cauchy Riemann differential operator $D$ on $\Gamma$:
$$D:\Gamma \...
- 207
4
votes
1
answer
247
views
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 ...
- 207
3
votes
0
answers
67
views
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 ...
- 956
6
votes
1
answer
746
views
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[...
- 61
7
votes
0
answers
571
views
Hochschild cohomology of a universal enveloping algebra of a Lie algebra
I was told that the following equation is true:
Given a finitely generated Lie algebra $\mathfrak g$, there is a Gerstenhaber algebra isomorphism
$$ HH(U\mathfrak g) \cong HH(\wedge^* \mathfrak g^\vee,...
- 323
21
votes
2
answers
3k
views
intuition for hochschild homology
According to this post Intuition for group homology, I wonder what is the intuition for Hochschild homology.
The Hochschild homology is defined as the homology of this complex
chain.
Given a ...
- 416
3
votes
1
answer
217
views
A differential graded Lie algebra with the Hochschild differential
Let $(V,\cdot)$ be an associative algebra and $W$ be a vector space endowed with a bimodule structure $\triangleright:V\otimes W\to W$ and $\triangleleft:W\otimes V\to W$ such that the following ...
- 629
6
votes
1
answer
355
views
Morita equivalence and isomorphisms in cohomology theories
Let $A,B$ be two unital algebras. We say that $A,B$ are Morita equivalent if there are $A-B$ and $B-A$ bimodules $P,Q$ such that
$$P \otimes_{B} Q \cong A, Q \otimes_A P \cong B$$
(as $A-A$ and $B-B$ ...
- 8,680
2
votes
1
answer
222
views
First Hochschild cohomology of $A=K[x]/(x^n)$
Given the algebra $A=K[x]/(x^n)$ for some field $K$ and natural number $n \geq 2$ with enveloping algebra $A^e=A \otimes_K A$.
It is easy to see that the 1. Hochschild cohomology of $A$ is nonzero ...
- 22.1k
6
votes
1
answer
329
views
Hochschild cohomology of certain local algebras
Let $K$ be a field and $A$ the algebra $K\langle x_1,...,x_n\rangle/J^m$ for $n \geq 1$ and $m \geq 2$, where $K\langle x_1,...,x_n\rangle$ is the non-commutative polynomial ring in $n$ variables over ...
- 22.1k
6
votes
1
answer
285
views
Lie algebra of a p-group
Given a p-group P, the first hochschild cohomology of the group algebra (over a field of characteristic p) of P is a nonzero Lie algebra. Is it known what Lie algebra results depending on P? I have no ...
- 22.1k
2
votes
0
answers
122
views
Another question on Hochschild cohomology
All algebras are assumed to be connected over a field.
Given an algebra $A$ with minimal faithful projective-injective left module $Af$ for some idempotent f.
Let $M$ be an $A$-module of infinite ...
- 22.1k
3
votes
1
answer
228
views
A question on Hochschild cohomology
Given a nonsemisimple symmetric algebra B and a non-selfinjective algebra A (all algebras are finite dimensional over a field and connected).
Can A and B have isomorphic Hochschild-cohomology rings?
- 22.1k
1
vote
0
answers
101
views
Hochschild coboundary on the space of alternative forms
Assume that $A$ is a complex algebra. By $C^{n}(A)$ we mean the space of all $n-$linear map $\phi:A^n \to \mathbb{C}$. An alternative $k-$ form is
an element $\phi \in C^{k}(A)$ ...
- 207
6
votes
0
answers
367
views
Definitions of Hochschild Cohomology $HH^{\bullet}(A)$
Let $A$ be an associative unital $k$-algebra, and let $M$ be a bimodule of $A$. The Hochschild cohomology of $A$ with coefficients in $M$ can be defined as
$$HH^{n}(A,\,M)=\mathrm{Ext}^{n}_{A^{e}}(A,\,...
- 631
9
votes
0
answers
518
views
From classical to quantum mechanics
Let ($X,\omega$) be a symplectic manifold (phase space of some physical system). Consider the algebra $\mathcal{C}^{\infty}(X,\mathbb{R})$ of smooth functions on $X$ and $[\omega]\in \textrm{H}^{2}_{\...
- 1,559
11
votes
0
answers
823
views
Is "Determinant" a Hochschild coboundary?
Assume that $n>2$.
Is there an associative unital algebra structure on $\mathbb{C}^{n}$ such that $D$, the determinant as a $n-\text{form} $ on $\mathbb{C}^{n}$,
would be a Hochschild ...
- 207
7
votes
1
answer
1k
views
What is the negative cyclic homology of a smooth projective variety?
Let X be a smooth and projective variety. Hochschild homology and cohomology have a very simple definition in terms of Ext groups of the diagonal of X. The Hochschild-Kostant-Rosenberg (HKR) theorem ...
- 1,829