Espaces de Berkovich globaux : cat\'egorie, topologie, cohomologie

Abstract

This text contributes to the foundations of the theory of global Berkovich spaces, that is to say Berkovich spaces over Banach rings with nice properties such as $\mathbf{Z}$, rings of integers of number fields, discrete valuation rings, hybrid fields, etc. We focus on three main themes that had not been investigated so far: category, topology and cohomology. As regards the category, our main task is to define a well-behaved notion of morphism. We then have the suitable setting at our disposal to carry out and study various constructions: products, fiber products, extensions of scalars, analytification of schemes, etc. On the topological side, we show that global Berkovich spaces are locally path-connected. The main ingredient is an analogue of Noether's normalization lemma, that we obtain after a careful study of finite morphisms. Finally, we prove that open and closed discs of arbitrary dimension have no higher coherent cohomology. This is a deep result, which allows us to initiate a theory of global overconvergent affinoid spaces, where the analogues of Tate's and Kiehl's theorems hold. As a consequence of our vanishing statements, we obtain a geometric proof of a Noetherianity result for certain rings of convergent arithmetic power series (power series with integral coefficients and positive complex radii of convergence), generalizing a theorem of D. Harbater from the case of a single variable to arbitrary many.