Rilind Hoti: Equivariant Riemann–Zariski spaces and flatification

Abstract: The Riemann–Zariski space of a variety X can be obtained both as the limit of blow-ups of X and as a space of valuation rings of the function field of X. In this paper, we study a proof of Raynaud and Gruson's flatification theorem due to Fujiwara and Kato that uses Riemann–Zariski spaces. By considering the limit of equivariant blow-ups of a scheme equipped with a group scheme action, we obtain a generalisation of the flatification theorem to an equivariant setting.