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!



Hochschild homology of dga's
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.


