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arXiv · 2507.08568

Some applications of the Nygaard filtration and quasisyntomic descent in positive characteristic

Livia Grammatica
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Abstract

This article gives an expository account of quasisyntomic descent and the Nygaard filtration in positive characteristic, complemented by several new applications to $p$-adic cohomology theories. The guiding result is a new approach to Illusie's comparison between fppf cohomology with $\mathbb{Z}_p(1)$ coefficients and the slope $1$ part of crystalline cohomology. We follow work of Bhatt-Lurie, but give a more elementary presentation which does not rely on the formalism of $\infty$-categories. We then revisit Ogus' comparison theorem between infinitesimal cohomology and \'etale cohomology, and give new proofs of several results on fppf cohomology that were previously obtained with the de Rham-Witt complex. We also determine the action of multiplication-by-$n$ on the fppf cohomology of an abelian variety, answering a question of A. Skorobogatov to the author. This is an expanded version of the author's master thesis.

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Livia Grammatica. 2025-07-11. Some applications of the Nygaard filtration and quasisyntomic descent in positive characteristic. https://arxiv.org/abs/2507.08568

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