Formal smoothness of the Artin-Mazur formal groups

Abstract

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 $\Phi^{i}(X,\mathbb{G}_m)$ 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 $\Phi^{i}(X,\mathbb{G}_m)$ 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 $\Phi^{i}(X,\mathbb{G}_m)$ is formally smooth when $i<d$, whereas $\Phi^{d}(X,\mathbb{G}_m)$ 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.