Question

Let $A$ be a semi-free commutative differential graded algebra in non-negative degrees with differential $d$ of degree $-1$, over a field of characteristic zero. Recall that semi-free means that if you forget the differential, then $A$ is free as a graded commutative algebra.

In Loday's book on cyclic homology, there is an explanation of how to compute Hochschild and cyclic homology of $A$ using a mixed complex built from a de Rham type complex for $A$ built from K\"ahler differentials and alternating powers of K\"ahler differentials. Is there a similar way to compute negative and periodic cyclic homology? The most naive thing to do is just to use the given mixed complex in the usual way, but that looks very large and it is not clear to me what the output would be.

A reference to such a computation would be very welcome.

Answer 1

Have you looked at rational homotopy techniques? If you want to compute the Hochschild homology of a cochain algebra $A$ (differential of degree +1), first you search a cofibrant replacement (a semi-free cdga) $F_A$ of $A$. Then you consider the multiplication $$m:F_A\otimes F_A\rightarrow F_A$$ this map admits a factorization $$F_A\otimes F_A\rightarrow (F_A\otimes F_A)\otimes_{\tau} W\rightarrow F_A$$ into a cofibration followed by a trivial fibration (a surjective quasi-isomorphism). The middle algebra is a resolution of $F_A$ as a $F_A$-bimodule (of course in the commutative case $F_A$-bimodules are $F_A\otimes F_A$-modules). Now you use the fact that Hochschild homology can be written as: $$Tor_{F_A\otimes F_A}(F_A,F_A).$$ This can be computed by a complex of the form: $$F_A\otimes_{F_A\otimes F_A}((F_A\otimes F_A)\otimes_{\tau}W)=F_A\otimes_{\tau}W.$$ You will find more details in "Rational homotopy theory" by Félix-Halperin-Thomas (Springer GTM 205) example 1 p206-207 where they give explicit formulas. Of course all is written in terms of cochain algebras and you use chain algebras, but I hope this will help you.

For cyclic homology Micheline Vigué-Poirrier has written papers on this topic using the preceding algebraic model.