# Hochschild homology of dga's

I'm sure the following statement is well-known to experts: Let $A$ be a dga. Let $perf(A)$ be the dg-category of perfect dg-modules over A. Then there is a quasi-isomorphism $$C_\bullet(perf(A)) \to C_\bullet(A)$$ between their Hochschild chain-complexes.

Does anyone know a reference for it?

I'm aware of Keller's 2003 paper that I think gives the result for Hochschild cochain-complexes, but I need the chain version. Thanks!

• I tried a brief look into the literature I know and only found the version for projective modules (McCarthy's "The cyclic homology of an exact category", Prop 2.4.3) - and if I delve any deeper I'd likely end up with a result proved for THH rather than HH which would require you to translate it. Sep 23, 2010 at 13:27
• Shklyarov, in Thm. 2.6 of his paper arxiv.org/abs/0710.1937 quotes a result saying that the natural map, which goes in the opposite direction of the one listed above, is a quasi-isomorphism. He references Keller's 1999 paper "On the cyclic homology of exact categories" for the proof, but gives no theorem number. While I don't doubt it's there somewhere, from a glance at Keller's paper I can't see where...I would love to see a concrete map if such a thing exists. Sep 23, 2010 at 20:05

I don't know a reference, but the assertion is easy to prove. The natural map goes in the opposite direction, $C_\bullet(A)\to C_\bullet(perf(A))$. It is quite simply induced by the embedding of the DG-category with a single object associated with $A$ into the DG-category $perf(A)$, sending the only object to the DG-module $A$ over $A$. To prove that it is a quasi-isomorphism, one can, e.g., interpret the Hochschild homology as the Tor of DG-bimodules, then use the fact that the derived categories of DG-bimodules over $A$ and $perf(A)$ are equivalent.
One can prove that what Caldararu-Tu call the Borel-Moore Hochschild homology are naturally isomorphic for a CDG-algebra $B$ and the DG-category $C$ of CDG-modules over $B$, projective and finitely generated as graded $B$-modules, in much the same way. I am finishing writing (or rather, editing now) a paper about this. It will be hopefully made public in a couple of weeks.