On Nash Blowing-Up

TL;DR: In this paper, the authors consider a sequence of transformations for algebraic algebraic variety X i over a base field k of characteristic zero, where X i is reduced and equidimensional, and they show that each transformation is locally free as the subsheaf consisting of those local sections whose supports are nowhere dense.

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 Kahler differentials on the variety and the torsion means the subsheaf consisting of those local sections whose supports are nowhere dense.