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

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 $Θ$-categories of Nuiten and Toën.

💡 Research Summary

The paper studies the Hochschild‑Kostant‑Rosenberg (HKR) spectral sequence for smooth varieties over a perfect field k of characteristic p > 0. The HKR theorem gives an isomorphism HH_i(R/k) ≅ Ω^i_{R/k} for a smooth k‑algebra R, and globally this yields a spectral sequence
E₂^{s,t}=H^s(X,Ω^{−t}{X/k}) ⇒ HH{−s−t}(X/k).
When the dimension of X is less than p the sequence degenerates, but in general it can have non‑trivial differentials. The main contributions are two theorems describing exactly which differentials can be non‑zero and how they are computed.

Theorem A has two parts. First, it proves that all differentials d_r with r < p vanish. This follows from the structure of the filtered circle S¹_fil, a deformation of the classifying stack of the Cartier dual of a formal group, which induces the HKR filtration. Second, assuming X admits a lift ˜X to the second Witt vectors W₂(k), the paper constructs a natural map
V : Ω¹_{X/k}