Formal smoothness of the Artin-Mazur formal groups

TL;DR

This work analyzes the Artin-Mazur formal groups Φ^{i}(X, G_m) of a smooth proper variety X in characteristic p, giving cohomological criteria for formal smoothness and linking these to crystalline and Witt vector cohomology. The authors introduce C-smoothness as a practical detector for formal smoothness and leverage Nygaard filtrations to connect fppf and crystalline invariants, obtaining a torsion-free sufficient condition and a torsion-based necessary condition. They construct Igusa-type, characteristic-2 examples showing that Φ^{i} can be formally smooth for i<d but fail at i=d, illustrating sharpness and guiding principles for non-smooth behavior. An equivalent formulation in terms of Witt vector cohomology provides a unifying perspective and a computable criterion, tying formal smoothness to the surjectivity of H^{i}(X, W) → H^{i}(X, O_X). The results deepen understanding of higher Artin–Mazur groups and offer concrete methods to diagnose and engineer formal smoothness via crystalline and Witt-theoretic data.

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 $Φ^{i}(X,\mathbb{G}_m)$ 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.

Figures (7)

• Figure 1: The region where the spectral sequences of $X$ and $\tilde{E}$ coincide
• Figure 2: the $E_2$ page
• Figure 3: the $E_3$ page
• Figure 4: the $E_2$ page
• Figure 5: the $E_3$ page

Theorems & Definitions (74)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Lemma 2.2
• proof
• Remark 2.3
• Proposition 2.4
• proof