2
$\begingroup$

In his paper Cyclic homology, Derivations and the Free Loopsace, Goodwillie defines periodic cyclic homology for differential graded algebras (A,d) concentrated in non-negative degree. Why does he make this restriction?

Elsewhere, for example in Jones' work on the same topic, periodic cyclic homology is defined for arbitrary dga's. Suppose that we have a dg-algebra which is co-connective, e.g. $A_n=0$ for $n>0$.

Is it true that $\operatorname{HP}_*((A,d)) \cong \operatorname{HP}_*(H_0(A))$ ?

$\endgroup$
2
  • $\begingroup$ No, this is not true. You can probably see this in any non-trivial example... $\endgroup$
    – naf
    Dec 10, 2014 at 5:56
  • $\begingroup$ I would like to guess that this is due to the fact that for all ring spectra $A$, both Postnikov sections $A\to\pi_0 A$ and surjections of discrete rings with nilpotent kernels are compositions of square-zero extensions. For the former being "square-zero", I would like to refer to Lurie's Higher Algebra Definition 7.4.1.18 of $n$-small extensions. $\endgroup$
    – user20948
    Sep 28, 2019 at 16:30

0

Your Answer

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