TITLE: Hochschild homology of dga's
QUESTION [8 upvotes]: 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!

REPLY [7 votes]: 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.