A New Proof of Raynaud–Gruson’s Flattening Theorem

Abstract

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 all 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.