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

TL;DR

This work analyzes the HKR spectral sequence in positive characteristic, proving $d_r=0$ for $r<p$ and giving a formula for the page-$p$ differential when a lift to $W_2(k)$ exists, expressed as $d_p=[V,\mathrm{Bock}_{\tilde X}]$ with $V$ a $p$th-power operation on the Atiyah class. The central technique uses the filtered circle of Moulinos–Robalo–Toën to model the HKR filtration as a deformation, computes the deformation class via a Bockstein attached to a lift, and then interprets $V$ as the Verschiebung acting on the Atiyah cobracket in a Tannakian and derived-coaffine setting. The paper then develops the Atiyah-class formalism and restricted-Lie-algebra structure in derived geometry, showing how $V$ recovers the classical $p$th-power operation on Lie algebras of group schemes and yields explicit computations for classifying stacks such as $B\mathbb G_m$, $B\mu_p$, and projective spaces. The results connect HKR degeneration phenomena to lift-torsion data, Adams operations, and partition-Lie structures, and point to broader extensions to $E$-HKR theories for one-dimensional formal groups and beyond.

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.

Theorems & Definitions (102)

• Theorem A
• Theorem B
• Definition 2.3.1
• Definition 3.1.1
• Definition 3.1.2
• Definition 3.1.3: ABM21, Definition 3.1(a)
• Definition 3.1.4
• Theorem 3.1.5
• Theorem 3.1.6
• Corollary 3.1.7