8
$\begingroup$

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!

Improve this question
$\endgroup$
2
  • $\begingroup$ 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. $\endgroup$
    – Tyler Lawson
    Sep 23, 2010 at 13:27
  • 1
    $\begingroup$ 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. $\endgroup$
    – Jesse Burke
    Sep 23, 2010 at 20:05

1 Answer 1

Reset to default
7
$\begingroup$

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.

Improve this answer
$\endgroup$
1
  • $\begingroup$ I'm writing up the same proof, as you probably suspected... Well actually I'm just correcting my existing (wrong) paper, using Caldararu-Tu's definition. I look forward to reading your version! $\endgroup$
    – Ed Segal
    Sep 24, 2010 at 8:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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