Related papers: Espaces de Berkovich sur $\mathbb{Z}$: morphismes …

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…

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,…

Synthetic algebraic geometry uses homotopy type theory extended with three axioms to develop algebraic geometry internal to a higher version of the Zariski topos. In this article we make no essential use of the higher structure and use…

This a first step to develop a theory of smooth, etale and unramified morphisms between noetherian formal schemes. Our main tool is the complete module of differentials, that is a coherent sheaf whenever the map of formal schemes is of…

In this paper we study relative Riemann-Zariski spaces attached to a morphism of schemes and generalizing the classical Riemann-Zariski space of a field. We prove that similarly to the classical RZ spaces, the relative ones can be described…

We prove that for any compact quasi-smooth strictly $k$-analytic space $X$ there exist a finite extension $l/k$ and a quasi-\'etale covering $X'\to X\otimes_kl$ such that $X'$ possesses a strictly semistable formal model. This extends a…

Topologies on algebraic and equational theories are used to define germ determined, near-point determined, and point determined rings of smooth functions, without requiring them to be finitely generated. It is proved, that any commutative…

We develop in this article flattening techniques for coherent sheaves in the realm of Berkovich spaces; we are inspired by the general strategy that Raynaud and Gruson have used for dealing with the analogous problem in scheme theory. As an…

We define normalized versions of Berkovich spaces over a trivially valued field $k$, obtained as quotients by the action of $\mathbb R_{>0}$ defined by rescaling semivaluations. We associate such a normalized space to any special formal…

In this paper, we first study the local rings of a Berkovich analytic space from the point of view of commutative algebra. We show that those rings are excellent ; we introduce the notion of a an analytically separable extension of…

The aim of this paper is to study the topological properties of algebraic sets with zero divisors. We impose a subbasic topology on the set of proper ideals of a $k$-algebra and this new ``$k$-space'' becomes a generalization of the…

Let $F$ be a field, let $D$ be a subring of $F$, and let ${\mathfrak{X}}$ be the Zariski-Riemann space of valuation rings containing $D$ and having quotient field $F$. We consider the Zariski, inverse and patch topologies on…

We define Liu morphisms and quasi-Liu morphisms between Berkovich analytic spaces. We show that Liu morphisms and quasi-Liu morphisms behave exactly as affine morphisms and quasi-affine morphisms of schemes.

Let $u:A\to B$ be a morphism of noetherian local rings. We obtain smoothness criteria for algebras with differential bases, in the case of rings containing a field of characteristic $p>0.$ We also give smoothness criteria for reduced…

It is known since the works of Zariski that the essential difficulty in the local uniformization problem is met already in the case of valuations of height one. In this paper we prove that local uniformization of schemes and non-archimedean…

In an attempt to classify all of the overlap-free morphisms constructively using the Latin-square morphism, we came across an interesting counterexample, the Leech square-free morphism. We generalize the combinatorial properties of the…

We investigate the topological properties of Berkovich analytifications over hybrid fields, that is a field equipped with the maximum of its native norm and the trivial norm. We prove that the analytification of the affine line or of a…

We consider a finite \'etale morphism $f:Y \to X$ of quasi-smooth Berkovich curves over a complete nonarchimedean non-trivially valued field $k$, assumed algebraically closed and of characteristic 0, and a skeleton…

Relations between some kinds of formal and standard smoothness, for morphisms of schemes, are clarified in surprisingly simple and direct ways, bypassing much of the customarily employed machinery. Even the deep local-to-global property of…

In this article, we give a full description of the topology of the one dimensional affine analytic space $\mathbb{A}_R^1$ over a complete valuation ring $R$ (i.e. a valuation ring with "real valued valuation" which is complete under the…