Equidimensionality of affine Deligne-Lusztig varieties in mixed characteristic

TL;DR

This paper proves that affine Deligne-Lusztig varieties in mixed characteristic are equidimensional, confirming a conjecture and extending previous results to all irreducible components through new geometric methods.

Contribution

It introduces a novel approach by translating Hartl-Viehmann's work into mixed characteristic and developing a formal algebraic geometry theory for perfect schemes.

Findings

Proves equidimensionality of affine Deligne-Lusztig varieties in mixed characteristic.

Extends results of Nie and Zhou-Zhu to all irreducible components.

Develops a theory of formal algebraic geometry for perfect schemes.

Abstract

We prove the equidimensionality of affine Deligne-Lusztig varieties in mixed characteristic. This verifies a conjecture made by Rapoport and implies that the results of Nie and Zhou-Zhu can be extended to the whole irreducible components of affine Deligne-Lusztig varieties. The method is to translate the work of Hartl-Viehmann into mixed characteristic and construct local foliations for affine Deligne-Lusztig varieties. This leads us to develop a theory of formal algebraic geometry for perfect schemes.