Reference for proof of Hochschild–Kostant–Rosenberg for Hochschild cohomology Is there a place where there is a full proof of the Hochschild–Kostant–Rosenberg theorem for Hochschild cohomology? I am aware of many places where the result is proven for Hochschild homology, i.e., Weibel, Ginzburg’s notes, etc. But I haven’t found a place where it’s proven for cohomology. Thanks!
 A: These notes might help, still the case considered by Kontsevich:
http://arxiv.org/abs/1107.0487
A: The case of smooth manifold is actually quite involved, since one need to define the tensor product of "topological convex algebras", which in general at least involves projective and injective tensor products. For the case of a compact manifold, it is proved by Gronthendieck that $C^{\infty}(M)$ is a nuclear space. Thus the two notions coincide and you can replace the norm completions with products of functions over $n+1$ copies of $M$. 
Having done that, proving Hochschild-Rosenberg-Konstant is now an execrise using derived functors by setting up the Mayer-Vietoris sequence and associated Cech-DeRham complex. The classical Poincare lemma then plays the same role as it does in the proof of DeRham's theorem, and the proof is almost obvious. However, this approach may not work in the algebraic setting in general as we did not use any Kahler differentials. So there is room to improve this proof. 
