The tag has no wiki summary.

learn more… | top users | synonyms

2
votes
0answers
53 views

Poincaré Duality of a quasi-free algebra

I'm completely stumped on this one (yet I feel it is obviously true or obviously false) If $A$ is a quasi-free algebra, then must it satisfy Poincaré duality? All i need to find is a protective ...
0
votes
1answer
86 views

Jacobi-Zariski exact sequence question

Denote by $HC(A,M)$ the Hochschild homological complex of an algebra $A$ with coefficients in an $A$-bimodule $M$, and let $B\rightarrow A$ be an $R$-flat extension of $R$-algebras, for some $CRing$ ...
1
vote
0answers
35 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
62 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
83 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
200 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 ...
2
votes
0answers
80 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
300 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
125 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
158 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
161 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
731 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 ...
4
votes
1answer
359 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
284 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
254 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, ...
11
votes
3answers
497 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
217 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
952 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
461 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
369 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
562 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) ...
5
votes
1answer
591 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
806 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
246 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
644 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
786 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
924 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 ...