The hochschild-homology tag has no wiki summary.
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
84 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
66 views
Hochschild Homology Subalgebra
MY question is simple, suppose $R$ is a $CRing$, and $A$ is an $R$-a subaglebra of the $R$-algebra $B$.
Then, how is the Hochschild homology of $A$ with values in the $A$ bimoduile $M$ related to ...
3
votes
1answer
190 views
Hochschild homology of quiver algebras
Let $K$ be a field and $\Gamma$ a quiver (=multidigraph) and $K[\Gamma]$ its quiver algebra (free $K$-module on the set of all paths of length $\geq0$ where multiplication is concatenation if ...
10
votes
2answers
393 views
Microlocalizing Hochschild homology
A recent paper of Ben-Zvi and Nadler gives a general formalism for understanding "dimensions" in sheaf theories. Without getting too far into details, amongst other things, this formalism allows us ...
5
votes
2answers
219 views
Hochschild homology of upper triangular matrix algebra?
Let $K$ be a field and $A$ the associative unital $K$-algebra of all $n\times n$ upper triangular matrices with entries in $K$. What is $\dim_K$ of its hochschild homology $HH_k(A;A)$?
Is there any ...
5
votes
1answer
84 views
Is there any relation between the simplicial $S^1$ and the Hochschild homology of a noncommutative algebras
Let $k$ be the base field and $A$ be a unital associative $k$-algebra. Let's review the Hochschild homology theory: we have the Hochschild chain comple $C_{\cdot}(A)$ where
$$
C_n(A):=A^{\otimes n+1}
...
3
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0answers
144 views
Hochschild homology of a tensor algebra modulo a two-sided ideal
Let $V$ is a module over a field $k$, and $A=T(V)$ the tensor algebra generated by $V$. The Hochschild homology $HH_*(A)$ has been determined by Loday and Quillen in their paper "Cyclic homology and ...
9
votes
2answers
729 views
Applications of topological chiral homology and factorization algebras (aka higher Hochschild cohomology)
I recently heard a talk about these topics and found them very interesting.
The talk was centered on the formal structure and didn't really focus on examples.
So my question is: what is your favorite ...
4
votes
1answer
207 views
Hochschild homology and change of non-ground ring
Let $k$ be a field, $R$ is a commutative algebra over $k$ and $A$ is an associative algebra over $R$. There is a morphism of commutative algebras $R \to T$. Is it possible to reduce calculation of ...
7
votes
1answer
359 views
Is there an algebraic “derived mapping space” construction that encompasses both Hochschild homology and loop spaces of non-simply-connected spaces?
I'm looking for directions to the literature that might contain fairly explicit constructions that might be called (the algebra of functions on) the "derived mapping space" from a simplicial set to a ...
3
votes
1answer
355 views
When do Hochschild homology and cohomology agree? (Ambidexterity?)
Suppose $X$ is a smooth algebraic variety over a field of characteristic $0$. What are the most general conditions under which Hochschild homology and cohomology of $X$ agree?
The existence of a ...
10
votes
1answer
379 views
Hochschild (co)homology of differential operators
I googled the title on the internet, and arrived at the following result
$$HH_k(D)\cong H_{DR}^{2n-k}(M).$$
Here $M$ is a smooth manifold of dimension $n$, and $D$ is the ring of differential ...
10
votes
3answers
504 views
Geometric realization of Hochschild complex
Let $A$ be a commutative $\mathbb{C}$-algebra, and consider $C_{\bullet}(A,A)$ the simplicial Hochschild homology module of $A$ with respect to itself (i.e. $C_{n}(A,A)=A^{\otimes (n+1)}$). This is a ...
5
votes
2answers
448 views
Does anyone recognize this quiver-with-relations?
Below I describe an infinite (but locally finite) quiver with relations. My question is whether anyone recognizes it and can provide appropriate pointers to the literature. I'm mainly interested in ...
7
votes
2answers
561 views
Relation between Gerstenhaber bracket and Connes differential
Let $C$ be an arbitrary algebra (more generally, a linear 1-category). The following structures are well-known:
A degree-0 product on the Hochschild cohomology $HH^*(C)$
$$
HH^*(C) \otimes HH^*(C) ...
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 ...
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 ...
8
votes
1answer
2k views
Hochschild (co)homology of Fukaya categories and (quantum) (co)homology
There is a conjecture of Kontsevich which states that Hochschild (co)homology of the Fukaya category of a compact symplectic manifold $X$ is the (co)homology of the manifold. (See page 18 of ...
12
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0answers
486 views
References for a certain generalization of Hochschild cohomology?
Let $C$ be an algebra. Let $E = C^{\otimes 2n}$ be the tensor product (over the ground field) of $2n$ copies of $C$. [EDIT: Or better, $E = C\otimes C^{op}\otimes C\otimes C^{op}\cdots\otimes C ...
5
votes
2answers
335 views
What is $TC(\Sigma^\infty \Omega X)$?
I know that for $X$ a connected space, $THH(\Sigma^\infty \Omega X) = \Sigma^\infty \Lambda X$, the suspension spectrum of the free loop space of $X$. The computation can be carried out in spaces and ...
10
votes
3answers
1k views
How exactly is Hochschild homology a monad homology?
Many texts which praise the generality of the bar construction associated to a monad, say that Hochschild homology is an example of this.
What exactly is in this case the underlying endofunctor of ...
3
votes
2answers
331 views
What is the homology of the real coordinate ring of SO(n,R)? Other compact matrix groups?
As someone whose knowledge of cohomology is patchy and picked up on a need-to-know basis, and whose algebraic geometry is even worse, I wondered if someone could help with this question. (I ran into ...
12
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4answers
1k views
Deligne's conjecture (the little discs operad one)
Deligne's conjecture states that the Hochschild cochain complex of an A-infinity algebra is an algebra over the operad of chains on the topological little discs operad.
Of course the conjecture has ...
6
votes
3answers
1k views
Definition of Hochschild (co)homology of a (dg or A-infinity) category
How do you define the Hochschild (co)homology of a dg category or an A-infinity category? I've only seen it defined when the category is equivalent to a category of modules over a dg algebra; then the ...
8
votes
5answers
808 views
Why is the Hochschild homology of k[t] just k[t] in degrees 0 and 1?
Background: the Hochschild homology of an associative algebra is the homology of the complex
... --> A (x) A (x) A --> A (x) A --> A
where the last two ...
