Stratifications of schemes using tangent vector fields

Abstract

Let $X/k$ be a noetherian scheme over a field $k$ of characteristic 0, such that the residue field at its closed points are algebraic extensions of $k$. Let ${\mathfrak{g}}_{X/k}\subset T_{X/k}$ be an ${\mathcal{O}}_X$-submodule of the tangent sheaf which is closed under the Lie bracket. We construct a stratification of the subset of closed points in $X$ by locally closed subsets that are preserved by ${\mathfrak{g}}_{X/k}$, on which ${\mathfrak{g}}_{X/k}$ acts transitively.

There are simple counterexamples to Lemma 1.1, so there may not exist any "defining points"