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

TL;DR

This paper explores quasisyntomic descent and the Nygaard filtration in positive characteristic, providing new applications to p-adic cohomology, simplifying existing proofs, and answering open questions about cohomology actions.

Contribution

It offers a more elementary approach to key comparisons in p-adic cohomology and introduces new proofs and applications in the field.

Findings

New approach to Illusie's comparison theorem

Elementary proofs of Ogus' comparison theorem

Determination of multiplication-by-n action on fppf cohomology

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.