All Questions
Tagged with hochschild-homology homological-algebra
31 questions
4
votes
1
answer
469
views
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://...
- 959
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)) =...
- 91
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 $A$ ...
- 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:=...
- 609
2
votes
1
answer
450
views
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,...
- 9,019
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 ...
- 26.5k
7
votes
0
answers
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) ...
- 1,144
3
votes
1
answer
280
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 ...
- 177
5
votes
1
answer
368
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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 ...
user39380
9
votes
0
answers
198
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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 ...
- 9,019
9
votes
1
answer
593
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 ...
- 1,382
5
votes
1
answer
685
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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 ...
- 1,382
2
votes
1
answer
169
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 ...
- 7,972
21
votes
2
answers
4k
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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 ...
- 416
7
votes
0
answers
416
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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,\,...
- 661
7
votes
1
answer
2k
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 ...
- 1,889
4
votes
0
answers
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 $(...
- 9,340
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 $A$ ...
- 6,259
2
votes
1
answer
578
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 ...
- 5,405
1
vote
0
answers
213
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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?
- 11
1
vote
1
answer
274
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 $CRing$ $...
- 5,405
6
votes
1
answer
1k
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 ...
- 1,589
8
votes
2
answers
721
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 ...
- 1,589
11
votes
1
answer
387
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
$$
C_n(A):=A^{\otimes n+1}
$...
- 9,019
3
votes
0
answers
300
views
Hochschild homology of a tensor algebra modulo a two-sided ideal
Let $V$ is a module over a field $k$, and $A=T(V)$ the tensor algebra generated by $V$. The Hochschild homology $HH_*(A)$ has been determined by Loday and Quillen in their paper "Cyclic homology and ...
- 31
5
votes
1
answer
392
views
Hochschild homology and change of non-ground ring
Let $k$ be a field, $R$ is a commutative algebra over $k$ and $A$ is an associative algebra over $R$. There is a morphism of commutative algebras $R \to T$. Is it possible to reduce calculation of ...
- 1,545
17
votes
1
answer
2k
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Is there an algebraic "derived mapping space" construction that encompasses both Hochschild homology and loop spaces of non-simply-connected spaces?
I'm looking for directions to the literature that might contain fairly explicit constructions that might be called (the algebra of functions on) the "derived mapping space" from a simplicial set to a ...
- 54.6k
11
votes
2
answers
1k
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Relation between Gerstenhaber bracket and Connes differential
Let $C$ be an arbitrary algebra (more generally, a linear 1-category). The following structures are well-known:
A degree-0 product on the Hochschild cohomology $HH^*(C)$
$$
HH^*(C) \otimes HH^*(C) \...
- 12.8k
7
votes
1
answer
408
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 ...
- 33.8k
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 \...
- 12.8k
16
votes
5
answers
2k
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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 ...
- 7,800