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

TL;DR

This paper extends the Hochschild-Kostant-Rosenberg decomposition theorem to smooth proper schemes in characteristic p when the dimension is less than or equal to p, using homological properties of Hochschild homology.

Contribution

It generalizes previous results by proving the HKR decomposition for dimension p, leveraging the self-duality of Hochschild homology.

Findings

HKR decomposition holds for dim X ≤ p in characteristic p

Hochschild homology of smooth proper schemes is self-dual

Extension from previous dim X < p result

Abstract

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.