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 a dg-category and hence has a Hochschild homology $HH_*(\mathbf{C}_A)$.

What is known about $HH_*(\mathbf{C}_A)$ for instance in low degree? How does it relate to the traditional Hochschild homology $HH_*(A)$ of the algebra $A$?

Note that this question is related to


but still different.

Thank you for any hints.


1 Answer 1


This is merely a couple of examples showing how $HH_*(\mathbf{C}_A)$ may behave.

Let first $A$ be the polynomial algebra $\Bbb C[x]$. Then $\mathbf{C}_A$ is the category of coherent sheaves with zero-dimensional support on $\Bbb A^1$. Since sheaves with supports on different points are orthogonal,this category is just the direct sum over all closed points of $\Bbb A^1$ of the corresponding categories, $\mathbf{C}_A = \oplus_{\lambda \in \Bbb C}~ \mathrm{Coh}_{\lambda}(\Bbb A^1).$

$Coh_\lambda(\Bbb A^1)$ is the category of vector spaces endowed with an endomorphism with all eigenvalues equal to $\lambda$. In particular, all these categories are equivalent to each other and, in particular, to $\mathrm{Coh}_{0}(\Bbb A^1)$, the category of vector spaces with nilpotent endomorphism, in other words, to $\mathbf{C}_{\Bbb C[[x]]}$, where $\Bbb C[[x]]$ is the power series algebra. Now, there's a localization sequence of triangulated categories:

$$\mathbf{C}_{\Bbb C[[x]]} \longrightarrow \Bbb C[[x]]-\mathrm{mod}_{\mathrm{fin. gen.}} \longrightarrow \Bbb C[x^{-1}, x]] - \mathrm{vect}_{\mathrm{fin. dim.}}$$

where $\Bbb C[x^{-1}, x]] - \mathrm{vect}_{\mathrm{fin. dim.}}$ is the category of finite dimensional vector spaces over the field of Laurent series. This sequence gives you a long exact sequence of Hochschild homology

$$\dots \longrightarrow HH_0(\mathbf{C}_{\Bbb C[[x]]}) \longrightarrow HH_0(\Bbb C[[x]]-\mathrm{mod}) \longrightarrow HH_0(\Bbb C[x^{-1}, x]] - \mathrm{vect}) \longrightarrow HH_{-1}(\mathbf{C}_{\Bbb C[[x]]}) \longrightarrow 0.$$

(see Keller, "On the cyclic homology of exact categories", https://www.sciencedirect.com/science/article/pii/S0022404997001527)

In particular, you can see that $HH_{-1}(\mathbf{C}_{\Bbb C[[x]]})$ is non-zero and is isomorphic to the polynomials in $x^{-1}$.

I don't know for sure, but I'd think that using something like adelic resolutions will show you that the category of sheaves with zero-dimensional support on a smooth variety of dim $n$ will have non-zero (in general) $HH_k$ for $k > -n$.

On the other hand, if you take $U{\mathfrak g}$, the universal enveloping algebra of some semisimple Lie algebra, then $\mathbf{C}_{U{\mathfrak g}}$ will be semisimple, so only $HH_0(\mathbf{C}_{U{\mathfrak g}})$ will be non-zero, while $HH_*(U\mathfrak g -\mathrm{mod})$ is isomorphic to the Lie algebra homology of $\mathfrak g$ with coefficients in the ring of invariant polynomials and is non-zero in arbitrary large degrees.

And of course if your algebra was finite dimensional from the beginning, you'll get nothing new.

  • $\begingroup$ Thank you for your examples. Does the last comment mean that $HH_*(\mathbf{C}_A)=HH_*(A)$ for finite-dimensional $A$? $\endgroup$ Jun 28, 2018 at 20:08
  • $\begingroup$ I mean that finitely generated modules are finite dimensional. So, $HH$ of finite dimensional modules is the same as $HH$ of finitely generated ones and is $HH$ of your algebra if every module is perfect. $\endgroup$ Jun 29, 2018 at 5:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.