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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

http://mathoverflow.net/questions/295876/hochschild-homology-of-a-hopf-algebra

but still different.

Thank you for any hints.

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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.

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  • $\begingroup$ Thank you for your examples. Does the last comment mean that $HH_*(\mathbf{C}_A)=HH_*(A)$ for finite-dimensional $A$? $\endgroup$ – Lukas Woike Jun 28 '18 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. $\endgroup$ – Grisha Papayanov Jun 29 '18 at 5:12

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