LULU is syntomic

Abstract

Let $G$ be a Chevalley group over a field $\Bbbk$. 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 \colon \mathbb{L} \times \mathbb{U}\times \mathbb{L} \times \mathbb{U} \longrightarrow G $ is syntomic and faithfully flat over any base field $ \Bbbk$.