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Questions tagged [hochschild-homology]

For questions about Hochschild homology of associative algebras and related concepts.

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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 ...
John Pardon's user avatar
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13 votes
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338 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 ...
Theo Johnson-Freyd's user avatar
12 votes
0 answers
552 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 \...
Kevin Walker's user avatar
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11 votes
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930 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 ...
Reimundo Heluani's user avatar
10 votes
0 answers
653 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 ...
Yuji Tachikawa's user avatar
9 votes
0 answers
366 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 ...
Mare's user avatar
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9 votes
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Does a morphism which induces an isomorphism between Hochschild homology also induce an isomorphism between cyclic homology?

In a 1998 paper by B. Keller, the author consider the following problem in Section 1.4: Let $k$ be a commutative ring and $X$ a scheme over $k$. We can consider the cyclic homology as well as the ...
Zhaoting Wei's user avatar
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8 votes
0 answers
481 views

Relationship between different definitions of the Hochschild homology

Throughout the literature, one can find many definitions of the Hochschild homology of various objects. However, the precise relationship between these definitions is not always so clear, at least to ...
Stahl's user avatar
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7 votes
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579 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) ...
Jake Wetlock's user avatar
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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,\,...
Yining Zhang's user avatar
6 votes
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263 views

A theory of higher limits of (1-)functors, after higher hochschild homology

$\newcommand{\Trans}{\mathrm{Trans}}\newcommand{\H}{\mathrm{H}}\newcommand{\HH}{\mathrm{HH}}\newcommand{\Hom}{\mathrm{Hom}}\newcommand{\Mod}{\mathrm{Mod}}$Recently I noticed that one may regard the co/...
Emily's user avatar
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6 votes
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121 views

Explicit homotopy for Hochschild chains from natural isomorphism

Let $A,B$ be $k$-linear (possibly, dg-)categories, let $f,g:A\to B$ be two linear functors, and let $T:f\Rightarrow g$ be a natural isomorphism. If one denotes by $C_\bullet(A,A)$ the standard ...
DamienC's user avatar
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249 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,...
Pedro's user avatar
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5 votes
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Two Hattori-Stallings trace questions

$\DeclareMathOperator\THH{THH}\DeclareMathOperator\Perf{Perf}\DeclareMathOperator\map{map}\DeclareMathOperator\tr{tr}\DeclareMathOperator\HH{HH}\DeclareMathOperator\fib{fib}\DeclareMathOperator\id{id}\...
Maxime Ramzi's user avatar
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Continuous and bornological Hochschild homology

As far as I understand, given a complex algebra $A$ with a locally convex topology $\mathcal{T}_A$ (e.g. $A=C^{\infty}(M, \mathbb C)$ for some manifold M), the topology induces a complete convex ...
Flavius Aetius's user avatar
5 votes
0 answers
87 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 $A_{L}$ ...
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4 votes
0 answers
317 views

What is the geometric interpretation of the first Hochschild homology group of path algebra constructed from a directed graph?

Let $\mathcal{G} = (V, E, s, t)$ is a directed graph, where $V$ - the set of its vertices, $E$ - the set of its edges, $s: E \rightarrow V, s((v_1, v_2)) = v_1$ and $t: E \rightarrow V, s((v_1, v_2)) =...
Alexander's user avatar
4 votes
0 answers
152 views

D-module theoretic Chern characters and an index-type theorem

Let $X$ be a smooth projective variety over $\mathbf{C}$. Consider the category $\mathbf{D}_{X}^{\text{perf}}$ of perfect complexes of left $D$-modules on $X$. It is well known that there is an ...
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4 votes
0 answers
158 views

Describing the THH of function spectra?

Are there any results describing the $THH$ of spectra of the form $F(X, E)$ where $X$ is a space (say, finite CW) and $E$ is a (nice enough) ring spectrum? I'm happy to put various (further, or ...
John Greenwood's user avatar
4 votes
0 answers
223 views

Deformation of the Hochschild-Kostant-Rosenberg isomorphism for universal enveloping algebra

Let $\mathfrak{g}$ be a Lie algebra over a char. $0$ field and let $\iota: U\mathfrak{g}\rightarrow S\mathfrak{g}$ be the Poincaré-Birkhoff-Witt (PBW) isomorphism, inverse to that natural map from the ...
user avatar
4 votes
0 answers
99 views

Computation of Hochschild homology

Let $A$ be a Dedekind domain. Let $n\geq 2$ be an integer. Is there a simple description of $HH_*(A, A/nA)$?
heygo's user avatar
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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}...
Just Me's user avatar
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4 votes
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211 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 $(...
truebaran's user avatar
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3 votes
0 answers
185 views

Hochschild homology of stable categories as topological chiral homology

Sorry for repost from Math Stack Exchange: Let $\mathscr{C}_0$ be a small idempotent complete stable category tensored over some symmetric monoidal category $\mathcal{E}$. Its Ind-completion $\mathscr{...
Chris Kuo's user avatar
  • 525
3 votes
0 answers
243 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 $A^{en}$-...
Mykola Pochekai's user avatar
3 votes
0 answers
169 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 ...
Libli's user avatar
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3 votes
0 answers
267 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 ...
math no more's user avatar
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3 votes
0 answers
300 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 ...
anonymous's user avatar
2 votes
0 answers
130 views

Deformation of Category via Hochschild Homology

Given a $\mathbb{C}$-linear category $\mathrm{C}$, let’s understand $\mathbf{HH}(\mathrm{C})$, the Hochschild homology of $\mathrm{C}$ as natural transformation. Then for any $A\in \mathbf{HH}(\mathrm{...
TaiatLyu's user avatar
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2 votes
0 answers
117 views

Spaces in the spectrum THH(R)

Let $R$ be a ring spectrum. Then we can form the topological Hochschild Homology of $R$ as the spectrum $$THH(R) = R \otimes S^1 \simeq R \wedge _{R \wedge R^{op}} R.$$ What is known about the spaces ...
categorically_stupid's user avatar
2 votes
0 answers
191 views

Poincare-duality for Hochschild Homology using Weibel's Hochschild sheaf

There is a notion of Poincare-duality for Hochschild homology, which works for $k$-algebras $A$ such that there is a $d\in \mathbf{N}$ with $\mathrm{Ext}^i(A,A^e)$ is zero except for $i=d$, and $\...
Aaron Wild's user avatar
2 votes
0 answers
140 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 $C$...
user avatar
2 votes
0 answers
90 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 ...
The confused man's user avatar
1 vote
0 answers
119 views

Cyclic homology with coefficients in a bimodule

I've recently been trying to understand Hochschild and cyclic co/homology better, and I've noticed that while it's common to define the Hochschild homology $\mathrm{HH}_{\bullet}(A;M)$ of an $R$-...
Emily's user avatar
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1 vote
0 answers
83 views

Hochschild homology computation of certain type

I know that in general Hochschild homology is not very computable. However, this certain type of Hoshschild homology shows up and I feel like there could be existing result. Let $k$ be a field and $A$ ...
Li Guanyu's user avatar
  • 449
1 vote
1 answer
329 views

Even and odd part of the Hochschild and cyclic homology of a super-algebra

Let $A$ be a $\mathbb Z_2$-graded $k$-algebra, where $k$ is a field of characteristic $0$. Then we know that the tensor product of $A$ with itself is also $\mathbb Z_2$-graded by $$(A\otimes_k A)_0:=...
Flavius Aetius's user avatar
1 vote
0 answers
50 views

Is there a version of Nest-Tsygan theorem for smooth variety

Let $M$ be a smooth Poisson manifold and $\mathcal{O}_\hbar (M)$ be a deformation quantisation of $\mathcal{O} (M)$. Nest-Tsygan theorem says that $$HH_i(\mathcal{O}_\hbar (M)[\hbar^{-1}])\cong H^{2d-...
FunctionOfX's user avatar
1 vote
0 answers
147 views

Simplicial realization of the circle action on the free loop space

Given a simply connected topological space $X$, it is well known that its free loop space $LX$ has cohomology being the Hochschild homology of the singular cochains [1]: $$HH_\bullet(S^\star X) \simeq ...
Student's user avatar
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1 vote
0 answers
60 views

Universal bimodule for homotopy biderivations

Recall the commutative story: for a commutative algebra $A$, its module of differentials $\Omega (A)$ is characterized by the universal property that any derivation $\delta \colon A \to M$ is in a ...
Dasha Poliakova's user avatar
1 vote
0 answers
64 views

How is the product structure induced on Lie algebra homology of matrices?

I have been looking at the Chevalley Eilenberg complex $CE_*(\mathfrak g)$ of a Lie algebra $\mathfrak g$ over a field $k$. $$ \wedge^3\mathfrak g\longrightarrow \wedge^2\mathfrak g\longrightarrow \...
Lies's user avatar
  • 11
1 vote
0 answers
120 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 $A$ ...
Bombyx mori's user avatar
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1 vote
0 answers
213 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?
Hunitaldude's user avatar