Questions tagged [hochschild-homology]
For questions about Hochschild homology of associative algebras and related concepts.
96 questions
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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{...
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How to calculate $\mathrm{TP}(\mathbb{F}_p[t])$?
$\DeclareMathOperator\TP{TP}$I am trying to learn about topological periodic cyclic homology following the notes:
https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/Papers/Lectures.pdf
https://...
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Is the exterior algebra intrinsically formal?
Following 4.6 and 4.7 of this paper by Seidel and Thomas, a graded algebra $A$ is called intrinsically formal if any two dgas with cohomology $A$ are quasi-isomorphic. There is a sufficient condition ...
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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 ...
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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 $R$-...
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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 ...
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An exact sequence involving THH
Disclaimer: I'm a relative beginner in this area. I'm trying to prove that if one has a commutative ring $R$ and a prime number $p$, then there is an exact sequence of the form
$$\DeclareMathOperator\...
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Even and odd part of the Hochschild and cyclic homology of a super-algebra
Let $A$ be a $\mathbb Z_2$-graded $k$-algebra, where $k$ is a field of characteristic $0$. Then we know that the tensor product of $A$ with itself is also $\mathbb Z_2$-graded by
$$(A\otimes_k A)_0:=...
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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)) =...
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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{...
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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/...
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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-...
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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 $A$ ...
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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 ...
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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 ...
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Can we compute the Hochschild cohomology for $k[x]$ through the Hochschild complex?
For an algebra $A$ we can define its Hochschild cohomology $HH^{\bullet}(A,A)$ as in this wikipedia page.
Now let $A=k[x]$ be the polynomial ring where $k$ is a field. It is well-known that $HH^{0}(A,...
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Strict graded commutativity of $\pi_*(\operatorname{THH}(A))$?
$\DeclareMathOperator\THH{THH}\DeclareMathOperator\HH{HH}$A version of the strict graded commutativity (i.e. graded commutativity & $x^2=0$ for every homogeneous element $x$ of odd degree) of $\...
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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 ...
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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 ...
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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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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}\...
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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 $g$ ...
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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 $X$...
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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 ...
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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 ...
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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
$$ \ldots \longrightarrow A \otimes A \otimes A \longrightarrow A \otimes A \longrightarrow A$$
where ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\...
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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) ...
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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)?
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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-...
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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 ...
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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 ...
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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 ...
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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 ...
user108998
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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 ...
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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 ...
user108998
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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