A new proof of Raynaud-Gruson's flattening theorem

TL;DR

This paper presents a new, direct proof of Raynaud-Gruson's flattening theorem using inverse limits of blow-ups, introducing a broader valuative space context that generalizes the original result.

Contribution

It provides a novel proof method and extends the flattening theorem to a wider class of valuative spaces.

Findings

New proof of Raynaud-Gruson's flattening theorem

Generalization to broader valuative spaces

Flattening property holds in the extended context

Abstract

We give a new proof of Raynaud-Gruson's theorem regarding flattening by blow-ups. The proof is direct, by working directly on the inverse limit of admissible blow-ups, which is a valuative space similar to the classical Zariski-Riemann space. These valuative spaces are defined in a broader context and always have an analogous flattening property: this yields a generalization of Raynaud-Gruson's theorem.