# Questions tagged [hochschild-homology]

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

91 questions
Filter by
Sorted by
Tagged with
567 views

### How to compute the periodic cyclic homology of this algebra

Let $k=\mathbb{C}$ be the field of complex numbers. I consider the (DG) algebra $A:=k[x]/(x^2)$ such that $\deg(x)=-1$. My question is how to compute the periodic cyclic homology, Hochschild homology ...
420 views

407 views

### What is $TP(\mathbb{Z}_p)$?

Let $TP$ be periodic topological cyclic homology. What is $\pi_* TP(\mathbb{Z}_p)$? (i) I know that $\pi_* TP(\mathbb{F}_p) \cong \mathbb{Z}_p[v^{\pm 1}]$ with $v$ in degree $-2$ by IV.4.8 of Nikolaus-...
359 views

### Defining Hochschild homology of non-commutative DG-algebras with animated rings with a circle action

A cool construction of Hochschild homology (that I saw on B. Antieau's website here ) is the following: Let $k$ be a commutative ring, then denote by $\mathfrak{a}\text{CAlg}_k$ the category of ...
242 views

283 views

### Morphisms of Hochschild (or cyclic) homology induced by homotopic maps

Let $f$ and $g$ be two maps between DG algebras $A$ and $B$, and assume that $f$ and $g$ are homotopic as chain maps, hence they induce the same map on the level of homology. Moreover, $f$ and $g$ ...
392 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 ...
505 views

### Measure the failure of colimit to commute with taking free loops (or Hochschild homology)?

For a space1 $X$, let $\mathcal{L}X = \mathrm{Maps}(S^1, X)$ be the free loop space. Inclusion of constant loops gives a natural map $X \to \mathcal{L}X$. This is not a homotopy equivalence unless $X$...
343 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 ...
491 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) ...
240 views

### Hochschild homology of acyclic complex

Let $A$ be a differential graded algebra over a commutative ring $R$. Suppose that $H_*(A)=0$, i.e. $A$ is acyclic. Question: Does this imply that the Hochschild homology $HH_*(A)$ also vanishes ...
1 vote
57 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 ...
148 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 ...
747 views

### How do you prove that Hochschild cohomology is Morita invariant?

I am simply trying to show that $HH^\bullet(A)= HH^\bullet(M_r(A))$ for any matrix ring of $A$. In Loday's book (Sect 1.5.6) the Morita invariance is explained as follows : it says that if $M$ is an ...
1 vote
52 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 \...
347 views

### Topological Hochschild homology of Azumaya algebra

Let $R$ be a commutative ring, let $A$ be an Azumaya algebra over $R$, does its topological Hochschild homology coincide with that of $R$? For example, let $\mathbb{H}$ be the quaternion algebra over ... 192 views

### 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 ...
110 views

### 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 ...
212 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 ... 98 views

### Computation of Hochschild homology

Let $A$ be a Dedekind domain. Let $n\geq 2$ be an integer. Is there a simple description of $HH_*(A, A/nA)$?
583 views

### Confusion about topological Hochschild homology and $\mathbb{Z}_p$-topological Hochschild homology

Let $R$ be the ring of integers in a perfectoid field of mixed characteristic $p$. Is $\pi_*THH(R)$ (as defined in Bhatt--Morrow--Scholze) $p$-complete (as an abelian group)?
396 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 ... 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 $A_{L}$ ... 387 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 ... 132 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 $C$... 2k 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 ...
243 views

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 $A\longmapsto A/[A,... 9 votes 1 answer 561 views ### Hochschild homology with coefficients in a certain bimodule Let$A$be a finite-dimensional$k$-algebra and$U$and$V$two finite-dimensional projective$A$-modules (maybe neither the finiteness nor projectivity has to play a role, but these requirements are ... 3 votes 0 answers 226 views ### Wrong way Poincare duality for Calabi-Yau dg-algebras? Let$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}$-... 8 votes 1 answer 368 views ### Algebraic models of non-simply connected spaces in string topology I want to find some algebraic models relating string topology to Hochschild and cyclic homologies. If the space$X$is simply-connected and we are working over rational numbers, we can use Sullivan ... 11 votes 0 answers 850 views ### Higher traces in Hochschild cohomology Let$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)$is a well ... 5 votes 1 answer 584 views ### Hochschild homology of a category of modules over an algebra Suppose$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$. This category can be seen as ... 3 votes 0 answers 151 views ### Hochschild homology and Chern character quiver with potential I am a beginner in quiver theory so this question might not be suitable for mathoverflow. Let$(Q,W)$be a quiver with potential and let$\Gamma$be the Ginzburg DG-algebra associated to$(Q,W)$. Is ... 2 votes 1 answer 153 views ### Variant of co-Tor in a bimodule category Say$\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}$). Then we have a notion ... 3 votes 0 answers 231 views ### Failure of periodic cyclic homology to be a localizing invariant A localizing invariant$E: dgcat_{sm} \rightarrow Ch_k$from small dg categories to chain complexes over a field$k$(say$k = \mathbb{C}$) is a dg functor which sends localization sequences to exact ... 4 votes 0 answers 102 views ### Moduli spaces for the TCFT map$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}...
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 ...