On the differentials of the Hochschild-Kostant-Rosenberg spectral sequence

TL;DR

This paper investigates the behavior of the Hochschild-Kostant-Rosenberg spectral sequence in positive characteristic, establishing conditions under which differentials vanish and providing explicit formulas involving Bockstein and Atiyah class operations.

Contribution

It demonstrates that differentials vanish before page p in characteristic p and derives a formula for the differential at page p when the variety lifts to W_2(k), involving new algebraic operations.

Findings

Differentials are zero before page p in characteristic p.

Provides a formula for the differential at page p involving Bockstein and Atiyah class.

Discusses Tannakian reconstruction for derived stacks.

Abstract

The Hochschild-Kostant-Rosenberg theorem implies the existence of a spectral sequence computing the Hochschild homology of a variety in terms of the cohomology of differential forms. When the base field $k$ has characteristic $p>0$, we show that the differentials in this spectral sequence are zero before page $p$; when the variety admits a lift to $W_2(k)$, we give a formula for the differential on page $p$. The formula involves the Bockstein associated to the lift and a $p$th power operation for the Atiyah class. Along the way, we also discuss rudiments of Tannakian reconstruction for derived stacks using the $\Theta$-categories of Nuiten and To\"en.