Let $A= \bigwedge \langle x_n \rangle $ be the free graded commutative algebra with differential zero ($x_n$'s are elements with positive grading )over a field of characteristic zero. I want a reference for computation of cohomology ring of $\bar{\mathrm{B}}^\bullet(A)$, reduced bar complex of $A$. It seems it's equal to $\bigwedge \langle x_n[1] \rangle $ but I can't do the computations.
Thanks!
It is done in the Cartan Seminar:
You can find the first part and the second part on Numdam. I think the precise reference is either the 3rd or 4th talk. If you do not read French or don't want to translate into modern language (they use the terminology of "multiplicative construction" of which the reduced bar complex is a special case), it follows from the Hochschild–Kostant–Rosenberg theorem, see e.g. Theorem 9.4.7 of
(because the homology of the reduced bar complex is the Hochschild homology with trivial coefficients $H_*(\bar{B}A) \cong HH_*(A; \Bbbk)$). The precise result is indeed that if $A$ is a symmetric graded algebra on generators $x_i$, then $HH_*(A;\Bbbk)$ is isomorphic as a graded vector space to the symmetric graded algebra on the generators $x_i[1]$ with degree shifted by $1$. Another possible reference is Proposition 6.2 of
though it is written in the language of operads; but I don't think there are major difficulties involved in recovering the original arguments for a plain algebra (or using $\mathsf{C}(\Sigma I) \circ V = S(\Sigma V) \xrightarrow{\sim} \operatorname{B}(\mathsf{C}(I)) \circ V \cong \operatorname{B}(S(V))$ because all the assumptions are satisfied).