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.