All Questions
9 questions
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
5
votes
1
answer
368
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 ...
user39380
5
votes
1
answer
685
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 ...
- 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
views
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
views
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
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?
- 11
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