Truncated Local Uniformization of Formal Integrable Differential Forms

We prove the existence of Local Uniformization for rational codimension one foliations along rational rank one valuations, in any ambient dimension. This result is consequence of the Truncated Local Uniformization of integrable formal differential 1-forms, that we also state and prove in the paper. Thanks to the truncated approach, we perform a classical inductive procedure, based both in the control of the Newton Polygon and in the possibility of avoiding accumulations of values, given by the existence of suitable Tschirnhausen transformations.

Quasiexcellence implies strong generation

We prove that the bounded derived category of coherent sheaves on a quasicompact separated quasiexcellent scheme of finite dimension has a strong generator in the sense of Bondal–Van den Bergh. This simultaneously extends two results of Iyengar–Takahashi and Neeman and is new even in the affine case. The main ingredient includes Gabber’s weak local uniformization theorem and the notions of boundedness and descendability of a morphism of schemes.