Mayer–Vietoris squares in algebraic geometry

Abstract

We consider various notions of Mayer–Vietoris squares in algebraic geometry. We use these to generalize a number of gluing and push out results of Moret-Bailly, Ferrand–Raynaud, Joyet and Bhatt. An important intermediate step is Gabber's rigidity theorem for henselian pairs, which our methods give a new proof of.