On Nash Blowing-Up

Abstract

Let X be an algebraic variety, reduced and equidimensional, over the base field k of characteristic zero. Let us consider a sequence of transformations

$$ {X_0} = X\xleftarrow{{{\sigma _1}}}{X_1}\xleftarrow{{{\sigma _2}}}{X_2} \leftarrow \cdots $$

where ${\sigma _i}:{X_i} \to {X_{i - 1}}$ for each $i \geqslant 1$ is

• (1) birational, i.e., proper and almost everywhere isomorphic, while X _i is reduced and equidimensional, and

• (2) $\sigma _i^*\left( {{\Omega _{{X_{i - 1}}}}} \right)$ /(its torsion) is locally free as ${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{O} _{Xi}}$ -module. Here Ω denotes the sheaf of Kähler differentials on the variety and the torsion means the subsheaf consisting of those local sections whose supports are nowhere dense.