Espaces de Berkovich sur Z : \'etude locale

TL;DR

This paper studies local properties of Berkovich spaces over Z, proving noetherianity, regularity, and coherence of structure sheaves, with methods applicable to various base rings and a unified approach for complex and p-adic spaces.

Contribution

It establishes foundational local properties of Berkovich spaces over Z and other rings, unifying complex and p-adic cases with new proofs.

Findings

Local rings of Berkovich spaces over Z are noetherian.

Affine spaces have regular local rings.

Structure sheaf is coherent.

Abstract

We investigate the local properties of Berkovich spaces over Z. Using Weierstrass theorems, we prove that the local rings of those spaces are noetherian, regular in the case of affine spaces and excellent. We also show that the structure sheaf is coherent. Our methods work over other base rings (valued fields, discrete valuation rings, rings of integers of number fields, etc.) and provide a unified treatment of complex and p-adic spaces.