The hochschild-cohomology tag has no wiki summary.
1
vote
0answers
31 views
Cohomology of a graded differential algebra with L-infinity action by a Lie algebra relative to a sub algebra
Suppose $A$ is a graded differential algebra, $h\subset g$ is an ideal, and that there is an $L_\infty$ action by $g/h$ on $A$. Is there any theorem that gives a quasi-isomorphism between the ...
3
votes
0answers
57 views
Lifting Lie algebra cohomology class to Hochschild cochain
Let $g$ be a Lie algebra, $h\subset g$ an ideal. The associative algebra $N=U(h)\subset U(g)$ can be viewed as a $g$-module.
The Lie algebra cohomology ${\rm H}^*(g,U(g))$ is isomorphic to the ...
1
vote
0answers
80 views
Sort of units for the Yoneda product (and/or in Hochschild cohomology)
In an abelian category $\mathcal A$ with enough projectives, we have the Yoneda pairing
$$\operatorname{Ext}^p_{\mathcal A}(Y,Z)\otimes \operatorname{Ext}_{\mathcal A}^q(X,Y)\longrightarrow ...
8
votes
0answers
192 views
A curious class in the Hochschild cohomology of graded algebras
If $A$ is a ($\mathbb Z$-)graded algebra, we can define its Hochschild cohomology in the usual way, via the standard complex of $A$-bimodules:
$$C_*(A)=\cdots\rightarrow A\otimes A\otimes ...
0
votes
0answers
59 views
$B_\infty$ algebra structure on $C(\mathcal{A}, \mathcal{A})$
If $A$ is an associative unital dg algebra it is known ( see for example: http://arxiv.org/abs/hep-th/9403055) that its Hochschild complex has the structure of a $B_\infty$ algebra. Given a dg ...
4
votes
2answers
286 views
Hochschild and cyclic cohomology of commutative algebra?
I'm wondering how to compute the Hochschild cohomology ${\rm HH}^n(A,A)$ and the cyclic cohomology ${\rm HC}^n(A)$ where $A = \mathbb{C}[t_i]$ is the algebra of polynomials in a finite set of ...
1
vote
0answers
119 views
Hochschild cohomology of de Rham algebra
I am interested in the relation between the Hochschild cohomology of the de Rham algebra on a manifold and the de Rham cohomology.
As we know, we can always take the de Rham algebra as a D.G.A. or ...
3
votes
1answer
154 views
The Hochschild cohomology of a variety “with coefficient” in a vector bundle
This question is related to one of my previous question Do we have the following isomorphism for $\mathcal{Ext}$?
Let $X$ be a smooth variety (over $\mathbb{C}$) and $\Delta: X \rightarrow X \times ...
3
votes
1answer
160 views
Is the definition of Gerstenhaber bracket related to operads?
I was reading these notes by Keller. On the page 19 he defines the Gerstenhaber bracket on the Hochschild cochain complex. But first of all he defines the operation $\bullet$ on the cochains by
$$
...
9
votes
2answers
685 views
Applications of topological chiral homology and factorization algebras (aka higher Hochschild cohomology)
I recently heard a talk about these topics and found them very interesting.
The talk was centered on the formal structure and didn't really focus on examples.
So my question is: what is your favorite ...
3
votes
1answer
349 views
When do Hochschild homology and cohomology agree? (Ambidexterity?)
Suppose $X$ is a smooth algebraic variety over a field of characteristic $0$. What are the most general conditions under which Hochschild homology and cohomology of $X$ agree?
The existence of a ...
0
votes
0answers
272 views
did any one know the definition of Hochschild cohomology of a differential graded algebra?
Let A with d be a differential graded associative algebra (DG algebra). What is the definition of Hochschild complex C*(A,A)? There should be a Hoshschild differential and a cup product, but i can't ...
7
votes
0answers
252 views
Algebras Morita equivalent to their centers
Hi,
I wonder if there is a name for:
1) Algebras which are Morita equivalent to their centers, or
2) dg-algebras which are derived Morita equivalent to their Hochshild cohomology?
For instance, ...
10
votes
3answers
476 views
Hochschild Cohomology of Differential Operators in characteristic 0
In Mariusz Wodzicki's paper "Cyclic homology of differential operators," the following result is mentioned: for $D_M$ the algebra of differential operators on a smooth manifold $M$ we have that ...
7
votes
0answers
216 views
Computer Algebra solution for simplicial resolutions for André-Quillen cohomology
Hello,
I would like to experiment with André-Quillen (co)homology. Especially for singular rings.
A key problem is that the construction of a simplicial resolution of a ring seems to require a rather ...
17
votes
4answers
943 views
A matrix algebra has no deformations?
I have often heard the slogan that "a matrix algebra has no deformations," and I am trying to understand precisely what that means. While I would be happy with more general statements about ...
2
votes
1answer
451 views
dg-lie structure on $HH^*$ and Koszul duality
This is shamelessly close to my other question: A Question on Koszul duality and $B(\infty)$ structures on $HH^*$. Maybe this one will get a better response. Rather than rewrite that one, I am going ...
4
votes
1answer
362 views
A Question on Koszul duality and $B(\infty)$ structures on $HH^*$
The following theorem is known from a paper "Duality in Gerstenhaber Algebras" by Felix, Menichi, Thomas. Given a simply connected space X of finite type.
There is an equivalence of Gerstenhaber ...
7
votes
2answers
554 views
Relation between Gerstenhaber bracket and Connes differential
Let $C$ be an arbitrary algebra (more generally, a linear 1-category). The following structures are well-known:
A degree-0 product on the Hochschild cohomology $HH^*(C)$
$$
HH^*(C) \otimes HH^*(C) ...
4
votes
1answer
561 views
A simple proof of the Weyl algebra's rigidity.
I am wondering if there is a nice presentation of the Hochschild cohomology of $A_n$ the Weyl algebra. It is known that $H^m(A_n,A_n)=0$ for $m>0$, and thus it is rigid. A proof can be found in ...
8
votes
1answer
788 views
What is the Hochschild cohomology of the dg category of perfect complexes on a variety?
Let $X$ be a quasi-projective variety over a field $k$. Let $D_{qcoh}$ be a dg enhancement of the unbounded derived category of quasi-coherent sheaves over $X$, and $D_{perf}$ its full subcategory of ...
6
votes
1answer
245 views
Hochschild H^1 (R,M) = 0 vs. H_1 (R,M) = 0 where R is a ring and M is an (R,R)-bimodule
Let $k$ be a commutative ring (with unity). Let $R$ be a $k$-algebra (with unity, but not of necessity commutative).
Let $M$ be an $\left(R,R\right)$-bimodule where $k$ acts in the same way from the ...
4
votes
3answers
1k views
Differential Hochschild Cohomology, general tools?
Background: in deformation quantization one wants to construct formal associative deformations (the star products $\star$) of the algebra of smooth complex-valued functions on a Poisson manifold $M$ ...
9
votes
2answers
639 views
Hochschild (co)homology of A and of Mod_A
Let A be an algebra (or dg algebra). Where can I find a proof of HH_*(A) = HH_*(Mod_A) and HH^*(A) = HH^*(Mod_A)? (And does this hold for any A?) Here Mod_A is, e.g., the category of left A-modules.
...
8
votes
1answer
784 views
“Spec” of graded rings?
From the discussion here, it seems that general Hochschild cohomology classes correspond to deformations where the deformation parameter can have nonzero degree.
So I have some naive and maybe stupid ...
3
votes
3answers
1k views
Hochschild cohomology and A-infinity deformations
When we are dealing with ordinary things or dg things (where thing = algebra or category), I think I understand how HH^2 corresponds to 1st order deformations and HH^3 corresponds to obstructions.
...
25
votes
5answers
3k views
Book on Hochschild (co)homology
There is currently no treatise treating Hochschild (co)homology systematically. There is a chapter in Weibel's book, there's parts of Loday's and a few others...
What should be covered by such a ...
11
votes
1answer
1k views
Is there a refinement of the Hochschild-Kostant-Rosenberg theorem for cohomology?
The HKR theorem for cohomology in characteristic zero says that if $R$ is a regular, commutative $k$ algebra ($char(k) = 0$) then a certain map $\bigwedge^* Der(R) \to CH^*(R,R)$ (where $\wedge^* ...
6
votes
0answers
908 views
Regarding the Gerstenhaber bracket on Hochschild cohomology and Morita equivalence
Associated to any $A_\infty$ $k$-algebra $A$ the Hochschild cochain complex $CH^*(A)$ has the structure of a dg-Lie algebra and a dg-algebra which are compatible enough that the cohomology is a ...
