Tate-linear Formal Varieties

Abstract

A Tate-linear structure on a smooth noetherian local formal scheme $T$ over a field $\kappa$ of characteristic $p$ is an isomorphism $T \xrightarrow{\sim} \mathbf{N}_{\mathbb{Q}}/\mathbf{N}$ of sheaves on the fpqc site of $\operatorname{Spec}(\kappa)$, where $\mathbf{N}$ is an fpqc sheaf of torsion free nilpotent on $\operatorname{Spec}(\kappa)$ which admits a central series $\mathbf{N} = \mathbf{N}_{1} \supsetneqq \mathbf{N}_{2} \supsetneqq \cdots \supsetneqq \mathbf{N}_{c+1} = (1)$ such that each subquotient $\mathbf{N}_{i}/\mathbf{N}_{i+1}$ is the Tate $\mathbb{Z}_{p}$-module attached to a $p$-divisible group over $\kappa$, and $\mathbf{N}_{\mathbb{Q}}$ is the Mal'cev completion of $\mathbf{N}$. A smooth formal scheme over $\kappa$ with a Tate-linear structure is called a Tate-linear formal variety over $\kappa$. Examples of Tate-linear formal varieties include $p$-divisible formal groups, biextensions of $p$-divisible formal groups, and formal completions at closed points of central leaves in Siegel modular varieties in characteristic $p$. Tate-linear structures have a remarkable rigidity property: if a reduced irreducible closed formal subscheme $W$ of a Tate linear formal variety $T$ is stable under the action of a group of Tate-linear automorphisms of $T$ which operates strongly nontrivially on $T$, then $W$ is a Tate-linear formal subvariety. Proofs of statements in this survey article can be found in Chapters 5–6 and 10–11 of [11].