A new proof of Raynaud-Gruson's flattening theorem

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.