arXiv subjects

Livia Grammatica

Publications and source records attributed to Livia Grammatica.

3 recordsLinked to original sources

arXiv2025

Brauer groups of abelian varieties over fields of finite characteristic

We study the Brauer group of an abelian variety A over an algebraically closed field of characteristic p focusing on the p-primary torsion, the key part of which is a certain quasi-algebraic unipotent group U_A. We determine its dimension and obtain a sharp upper bound for its p-exponent. The isogeny class of U_A is classified for abelian varieties A of dimension at most 3. For principally polarised abelian varieties we compute the dimension of the p-torsion subgroup of U_A in terms of the Ekedahl--Oort type of A.

math.AG↗

arXiv2025

Formal smoothness of the Artin-Mazur formal groups

Let $X$ be a smooth proper variety over an algebraically closed field of positive characteristic $p$. We find cohomological conditions for the Artin-Mazur formal group functors $Φ^{i}(X,\mathbb{G}_m)$ to be formally smooth. We show that if all crystalline cohomology groups of $X$ are torsion-free (e.g. if $X$ is an abelian variety) then all of the $Φ^{i}(X,\mathbb{G}_m)$ are representable and formally smooth. We then identify a necessary condition for formal smoothness, which we use to give examples, for any $d\ge2$, of varieties $X$ for which $Φ^{i}(X,\mathbb{G}_m)$ is formally smooth when $i<d$, whereas $Φ^{d}(X,\mathbb{G}_m)$ is not. The constructions are inspired by Igusa's surface with non-smooth Picard scheme. Finally, we give a condition equivalent to formal smoothness in terms of Serre's Witt vector cohomology. The strategy relies on the notion of $C$-smoothness - where $C$ is the group algebra of $\mathbb{Q}_p/\mathbb{Z}_p$ - which is a condition that detects when a formal group is formally smooth, and on the use of the Nygaard filtration to relate fppf cohomology to crystalline cohomology.

math.AG↗

arXiv2025

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

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 étale 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.

math.AG↗