For questions relating to the calculation or definition of Hochschild (co)homology, an algebraic invariant of associative algebras, dg algebras and dg categories.
Questions tagged [hochschild-cohomology]
57 questions
14
votes
1 answer
Hochschild homology - motivation and examples
I'm currently trying to learn about Hochschild homology of differential graded algebras. After reading the definition, the notion of Hochschild homology is somewhat unmotivated and myterious to me. What is the motivation to define Hochschild…
Dave
- 1,853
12
votes
0 answers
Many definitions of Hochschild homology and cyclic homology
It appears that there are more definitions of cyclic homology than there are people working on cyclic homology. As a newcomer, this confuses me to no end. I've written a list of definitions that some Google searches have given me, although I am…
nagger
- 141
12
votes
1 answer
Hochschild homology of Weyl algebra
Could someone explain to me how one can compute the Hochschild homology of the Weyl algebra $A_n$ (i.e., algebra of differential operators with polynomial coefficients in $n$ variables)?
Alex
- 6,684
10
votes
0 answers
Morita-invariance of Hochschild (co)homology.
Ok, I’m reading the paper Homology and cohomology of associative algebras. A concise introduction to cyclic homology by Christian Kassel, and on page 19 he says that Hochschild homology is Morita-invariant, by which he means that if $R$ and $S$ are…
L-A
- 367
10
votes
1 answer
Correct definition of Hochschild homology
In many expositions and discussions of Hochschild homology, it is stated that the classical definition via the cyclic bar complex (with underived tensor products) is incorrect (call the obtained Hochschild homology $\mathrm{HH}^{\textrm{cl}}$). To…
Stahl
- 23,855
8
votes
2 answers
Definition of Hochschild homology in terms of Tor functor (bar resolutions)
I had 2 kind of dumb questions about the definition of Hochschild homology in terms of the Tor functor:
1 - Let $R$ be a $k$-algebra and $M$ an $R$-bimodule, let $H_*(R,M)$ be the Hochschild homology of $R$ with coefficients in $M$, this homology…
The K
- 365
8
votes
1 answer
Computing Hochschild cohomology of an algebra in GAP
I would like to calculate Hochschild cohomology of a path algebra of a quiver (modulo some relations) using GAP.
Creating the quiver and its path algebra modulo relations is explained in detail in the QPA manual.
After having defined an algebra $A$…
Earthliŋ
- 2,592
7
votes
2 answers
Cap product for Hochschild (co)chains
Let $A$ be an associative algebra and $M$ be an $A$-bimodule. Then we can form the Hochschild cochains $C^\bullet(A,A)$ and chains $C_\bullet(A, M)$ and define a pairing (cap product)
$$
C^\bullet(A,A) \otimes C_\bullet(A, M)
…
C_M
- 731
6
votes
0 answers
Computing relative Hochschild (co)homology
I'm trying to compute/find simple examples the relative Hochschild cohomology of an extension of algebras. Let $$B \subseteq A$$ be an extension of algebras. Firstly there is the relative bar complex which Hochschild defined in his original…
Najonathan
- 519
6
votes
0 answers
Relation between Hochschild homology and cohomology
Let $A$ be an associative algebra, then we have the Hochschild chain complex, namely
$$
\dotsb \to A^{\otimes 3} \xrightarrow{d_2} A^{\otimes 2} \xrightarrow{d_1} A \,,
$$
where, for example, $d_1 (a \otimes b) = ab - ba.$ I want to apply…
iou
- 1,216
6
votes
1 answer
Wedge product of Hochschild cohomology classes in characteristic $2$
Let $A$ be a smooth commutative $k$-algebra, for $k$ a commutative ring. By the Hochschild–Kostant–Rosenberg theorem, we have that $\mathrm{HH}^*_k(A) \cong \bigwedge^* \mathrm{Der}_k(A, A)$, where $\mathrm{Der}_k(A,A)$ is the $A$-module of…
KReiser
- 74,746
6
votes
0 answers
Deformations of associative algebras and Hochschild cohomology.
I am studying the deformation theory of associative algebras (and Poisson algebras) and came across a question for which I cannot find an answer:
Let $(A, \mu)$ be a commutative associative algebra over a field $\mathbb{F}$. The first order…
unknownymous
- 1,063
5
votes
0 answers
Has this variation of Hochschild cohomology been studied?
Let $k$ be a field, and let $A$ be a commutative $k$-algebra.
Let $M$ be an abelian group, and assume that it an $n$-$A$-module. That is: it has $n$ different $A$-module structures, and they are pairwise compatible. (For example, if $M_1,\dots,M_n$…
user147705
- 51
5
votes
0 answers
Base-change for Hochschild Cohomology?
Let $f: R \to S$ be a homomorphism of commutative rings. If helpful, one can assume that $R$ is a subring of $S$. Now, let $A$ be an $S$-algebra and $M$ an $A$-bimodule. The Hochschild cohomology relative to $S$ is defined as the…
Chris Kuo
- 1,849
- 11
- 14
5
votes
3 answers
What is the connection between Grothendieck's Differential Operators and Hochschild Cohomology
For a given commutative algebra $A$ over a field $\mathbb{K}$(with char=0) the algebra of differential operators on $A$ is the set of endomorphism $D$ of $A$ such for some $n$ we have that for any sequence $\left\lbrace a_i\right\rbrace_{1\leq i\leq…
BBischof
- 5,967