Questions tagged [hochschild-homology]
For questions about Hochschild homology of associative algebras and related concepts.
3
votes
1answer
179 views
Hochschild Homology and Formal Geometry
My question concerns the degeneration of the spectral sequence computing Hochschild homology of differential operators on a smooth affine variety $X$.
The spectral sequence arises from the ...
5
votes
0answers
60 views
Existence of anti-symmetric hochschild homology representatives
Let $A$ be an associative algebra over a field $k$. Let $A_{L}$ be the Lie algebra of $A$ with commutator bracket. Then if $M$ is a bimodule for $A$ there is an associated representation of $A_{L}$ ...
5
votes
1answer
230 views
Confusion on summand of Hochschild homology of D-modules
I've encountered the following confusion.
Start with the category of perfect D-modules on $\mathbb{A} ^{1}$, which I'll denote $D$. We have the object $\mathcal{O}$, which is exceptional in the sense ...
2
votes
0answers
70 views
Computing Hochschild Invariants of Positselski's Coderived Categories
Positselski's work allows one to frame Koszul duality very elegantly in terms of so called coderived categories of modules over coalgebras, these are somewhat exotic dg categories of comodules over $C$...
15
votes
2answers
580 views
revisiting $THH(\mathbb{F}_p)$
Reading through Bhatt-Morrow-Scholze's "Topological Hochschild Homology and Integral p-adic Hodge Theory" I encountered the following statement.
We use only “formal” properties of THH throughout ...
5
votes
0answers
178 views
$\mathrm{HH}$ and $\mathrm{HC}$ as two different Taylor expansions at the same point
Let us consider the functors from dg algebras to graded spaces given by Hochshcild homology and cyclic homology. Then it is well known that in degree $0$, both functors are given by $A\longmapsto A/[A,...
9
votes
1answer
353 views
Hochschild homology with coefficients in a certain bimodule
Let $A$ be a finite-dimensional $k$-algebra and $U$ and $V$ two finite-dimensional projective $A$-modules (maybe neither the finiteness nor projectivity has to play a role, but these requirements are ...
3
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0answers
155 views
Wrong way Poincare duality for Calabi-Yau dg-algebras?
Let $A$ be a smooth compact Calabi-Yau dg $k$-algebra of dimension $w$. It is widely known (e.g. Atsusi Takahashi proposition 2.4) that in such situation we have non canonical isomorphism of $A^{en}$-...
5
votes
1answer
217 views
Algebraic models of non-simply connected spaces in string topology
I want to find some algebraic models relating string topology to Hochschild and cyclic homologies. If the space $X$ is simply-connected and we are working over rational numbers, we can use Sullivan ...
9
votes
0answers
379 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 ...
4
votes
1answer
216 views
Hochschild homology of a category of modules over an algebra
Suppose $A$ is an algebra over some field, say the complex numbers if that helps. Then we can consider the category $\mathbf{C}_A$ of finite-dimensional modules over $A$.
This category can be seen as ...
3
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0answers
92 views
Hochschild homology and Chern character quiver with potential
I am a beginner in quiver theory so this question might not be suitable for mathoverflow.
Let $(Q,W)$ be a quiver with potential and let $\Gamma$ be the Ginzburg DG-algebra associated to $(Q,W)$. Is ...
2
votes
1answer
73 views
Variant of co-Tor in a bimodule category
Say $\mathcal{C}$ is a strict monoidal abelian category and $A$ is a coalgebra object in $\mathcal{C}$, with left co-modules $M$ and right co-module $N$ (also in $\mathcal{C}$). Then we have a notion ...
2
votes
0answers
75 views
Failure of periodic cyclic homology to be a localizing invariant
A localizing invariant $E: dgcat_{sm} \rightarrow Ch_k$ from small dg categories to chain complexes over a field $k$ (say $k = \mathbb{C}$) is a dg functor which sends localization sequences to exact ...
3
votes
0answers
77 views
Moduli spaces for the TCFT map $HH(L) \to GW(X)$
Let $L$ be a Lagrangian submanifold of a closed symplectic manifold $X$. What I gather from Costello (see specifically $\S$2.5 there), is that one expects to have a morphism of closed TCFT's
$\tag{1}...
18
votes
2answers
1k 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 ...
-1
votes
0answers
163 views
Is Hochschild homology invariant under A-infinity quasi isomorphism?
If A and B are two A-infinity algebra, A is A-infinity quai-isomorphic to B. Do we have HH(A)=HH(B)?
9
votes
1answer
255 views
String cobracket from TFT
Let $M$ be a closed oriented manifold. Chas and Sullivan (https://arxiv.org/abs/math/0212358) introduced a Lie bialgebra structure on $H_\bullet^{S^1}(LM, M)$, $S^1$-equivariant homology of the loop ...
14
votes
1answer
696 views
Categorification of Floer homology
Floer homology associates a vector space $HF^\ast(L_1,L_2)$ to any pair of Lagrangian submanifolds $L_1,L_2$ inside a symplectic manifold $X$. By a categorification of Floer homology, I mean a ...
5
votes
1answer
212 views
Slick construction of Hochschild complex
Let $R$ be a $k$-algebra and $M$ be an $(R,R)$-bimodule. Let $[n] \mapsto M \otimes R^{\otimes n}$ be the simplicial $k$-module which defines the Hochschild homology $H_*(R,M)$. Is it possible to ...
1
vote
1answer
173 views
Does rational surface have exceptional collection of maximal length but not full?
Let $X$ be a rational surface. Let $\mathbb{E}:=(E_1,\ldots,E_n)$ be strong exceptional collection of line bundles of maximal length $l=rk Pic(X)+2$ in $D^b(coh(X))$, Is there any example that such ...
7
votes
1answer
225 views
Topological Hochschild homology and Hochschild homology of dg algebras
Topological Hochschild homology is a generalization of Hochschild homology from rings to $E_\infty$-ring spectra. On the other hand, there is a natural way to extend the notion of Hochschild homology ...
5
votes
0answers
304 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,\,...
7
votes
1answer
840 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 ...
6
votes
1answer
293 views
A mysterious quasi-isomorphism in Kashiwara-Schapira's proof of HKR
On p. 127 of Kashiwara-Schapira's paper "Deformation Quantization Modules", there is the following situation: $X$ is a smooth complex (quasi?)projective variety and $\delta\colon X\to X\times X$ is ...
4
votes
0answers
186 views
Inner automorphisms acts as identity on Hochschild homology
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 $(...
1
vote
0answers
86 views
Tor functor in the case of algebra of smooth functions
Let $A=C^{\infty}(\mathbb{S}^{1})$, let $B$ be the sub-algebra $C^{\infty}(0,1)$. Here we identify $\mathbb{S}^{1}$ by $\mathbb{R}/2\pi \mathbb{Z}$. I want to ask if there is a way I can decompose $A$ ...
10
votes
0answers
196 views
When does Hochschild homology commute with infinite products?
Let $A$ be an associative algebra. Its zeroth Hochschild homology $\mathrm{HH}_0(A)$ is the cokernel of the linear map $A^{\wedge 2} \to A$, $a \wedge b \mapsto ab - ba$. I.e. you quotient the ...
11
votes
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 ...
2
votes
1answer
376 views
Interpretation of Hochschild Homology groups
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 ...
4
votes
1answer
140 views
Hochschild chain model for the evaluation map at half
Let M be a manifold and $\Lambda(M)$ its free loop space, $A= C^*(M)$ denotes the cochain algebra of $M$. We know that Hochschild chain model for the evaluation $ ev_0: \Lambda(M) \rightarrow M$ is ...
1
vote
1answer
419 views
Hochschild cohomology and formal smoothness
Hochschild cohomology can be used to characterise formal smoothness of unital associative algebras; in that such an algebra $A$ is formally smooth if and only if it is of Hochschild cohomological ...
1
vote
0answers
129 views
Example H-unital algebra which is not unital
What is an example of an algebra which is H-unital (that is its Bar complex is acyclic) and yet it is not unital?
2
votes
0answers
81 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 ...
1
vote
1answer
180 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$ $...
5
votes
1answer
773 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 ...
12
votes
2answers
870 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 ...
8
votes
2answers
473 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 ...
10
votes
1answer
227 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
votes
0answers
242 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 ...
12
votes
2answers
2k 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 ...
5
votes
1answer
285 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 ...
13
votes
1answer
989 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 ...
5
votes
1answer
511 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
0answers
535 views
On cyclic homology of Ginzburg's DG algebra
Let $(Q,W)$ be a quiver with potential, and $D$ be Ginzburg's DG algebra associated to it (as explained in http://arxiv.org/abs/math/0612139 and other places), so that it is a 3-Calabi-Yau algebra. I ...
12
votes
1answer
534 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
637 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 ...
6
votes
2answers
664 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 ...
10
votes
2answers
897 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) \...
7
votes
1answer
328 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 ...
