Geir Ellingsrud (UiO) - New proofs of Gruson and Raynaud's flattening theorem using relative Zariski-Riemann spaces

Around 1970, Gruson and Raynaud proved that one may render a morphism of finite presentation flat by blowing up an ideal in the target, with rather mild restrictions on the involved schemes. New proofs, entirely different from the one of Gruson and Raynaud, have been found by Fujiwara and Kato and by Guignard. The two proofs are basically the same, both using Temkin's relative Zariski–Riemann spaces, the limits of all interesting blow-ups. These spaces are quasi compact locally ringed spaces whose local rings are close to being valuation rings, so that almost anything is flat, and due to their quasi compactness, one may descend flatness to a genuine blow-up.

Notes for the talk here.

References
Quentin Guignard: A New Proof of Raynaud–Gruson’s Flattening Theorem (arXiv:1801.01188v3)
Kazuhiro Fujiwara and Fumiharu Kato: Rigid Geometry and Applications, in Advanced Studies in Pure Mathematics 45 (2006), Moduli Spaces and Arithmetic Geometry (Kyoto, 2004), pp. 327–386.