LULU is syntomic

Abstract

Let $G$ be a Chevalley group over a field $\mathbb{k}$. Fix a maximal torus $\mathbb{T}$ in $G$, along with opposite Borel subgroups $B$ and $B^{-}$ satisfying $\mathbb{T}=B\cap B^{-}$, and denote by $\mathbb{U}:=R_u(B)$ and $\mathbb{L}:=R_u(B^{-})$ their respective unipotent radicals. We prove that the multiplication map $\mu :\mathbb{L}\times \mathbb{U}\times \mathbb{L}\times \mathbb{U}⟶G$ is syntomic and faithfully flat over any base field $\mathbb{k}$.