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I am currently reading Cyclic Operads and Cyclic Homology by Getzler-Jones and have some confusions.

I am under the impression that given an (associative, say) algebra $A$ that an almost-free resolution (think cofibrant replacement) of $A$, which we think of as a differential-graded vector space concentrated in degree $0$, is given by the free associative algebra $F(Ass, A)=\bigoplus_{i\geq 1} A^{\otimes n}$. Explicitly, this is supposed to be a dg associative algebra which is weakly equivalent to $A$ and is free after forgetting the differential. This is implicitly used in the proof of Lemma 5.12 which allows for a conclusion of the proof that the complex denoted $CA_*(A)$ computes the cyclic homology of $A$.

However, I have no idea why $F(Ass, A)$ should be such an almost-free resolution. I would naively expect the differential on $F(Ass, A)$ to correspond to the one coming from the bar construction, which would give the desired conclusion. However, following the construction of the differential on a free algebra from Loday-Vallette section 6.3.11 it seems to give $0$. Am I missing something obvious here?

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  • $\begingroup$ Hm, perhaps it's okay to simply choose my almost-free resolution to be the free algebra with the differential which forces it to be acyclic. The underlying thing is certainly free and it's weakly equivalent to the thing we started with. $\endgroup$
    – JD1874
    Commented Aug 13 at 12:52
  • $\begingroup$ I cannot tell where, in the proof of Lemma 5.12, they would use that $F(\mathrm{Ass}, A)$ with some differential is a resolution of $A$. Intuitively this doesn't look right - in any case, it's concentrated in degree 0 if $A$ is, so it cannot even have a nontrivial differential... $\endgroup$
    – Najib Idrissi
    Commented Aug 29 at 14:39

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