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arXiv · 2410.01894

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

Joshua Mundinger
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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.

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Joshua Mundinger. 2024-10-02. On the differentials of the Hochschild-Kostant-Rosenberg spectral sequence. https://arxiv.org/abs/2410.01894

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