A remark on the Hochschild-Kostant-Rosenberg theorem in characteristic p

We prove a Hochschild-Kostant-Rosenberg decomposition theorem for smooth proper schemes $X$ in characteristic $p$ when $\dim X\leq p$. The best known previous result of this kind, due to Yekutieli, required $\dim X<p$. Yekutieli's result follows from the observation that the denominators appearing in the classical proof of HKR do not divide $p$ when $\dim X<p$. Our extension to $\dim X=p$ requires a homological fact: the Hochschild homology of a smooth proper scheme is self-dual.