About Hrushovski and Loeser's work on the homotopy type of Berkovich spaces

Those are the notes of the two talks I gave in april 2013 in St-John (US Virgin Islands) during the Simons Symposium on non-Archimedean and tropical geometry. They essentially consist of a survey of Hrushovski and Loeser's work on the homotopy type of Berkovich spaces; the last section explains how the author has used their work for studying pre-image of skeleta.Comment: 31 pages. This text will appear in the Proceedings Book of the Simons Symposium on non-Archimedean and tropical geometry (april 2013, US Virgin Islands). I've taken into account the remarks and suggestion of the referee

Triangulation et cohomologie \'{e}tale sur une courbe analytique

Let $k$ be a non-archimedean complete valued field and let X be a smooth Berkovich analytic $k$-curve. Let $F$ be a finite locally constant \'{e}tale sheaf on $k$ whose torsion is prime to the residue characteristic. We denote by $|X|$ the underlying topological space and by $\pi$ the canonical map from the \'{e}tale site to $|X|$. In this text we define a triangulation of $X$, we show that it always exists and use it to compute $H^{0}(|X|,R^{q}\pi\_{*}F)$ and $H^{1}(|X|,R^{q}\pi\_{*}F)$. If $X$ is the analytification of an algebraic curve we give sufficient conditions so that those groups are isomorphic to their algebraic counterparts ; if the cohomology of $k$ has a dualizing sheaf in some degree $d$ (e.g $k$ is $p$-adic, or $k=C((t))$) then we prove a duality theorem between $H^{0}(|X|,R^{q}\pi\_{*}F)$ and $H^{1}\_ {c}(|X|,R^{d+1}\pi\_{*}G)$ where $G$ is the tensor product of the dual sheaf of $F$ with the dualizing sheaf and the sheaf of $n$-th roots of unity

Les espaces de Berkovich sont excellents

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 non-archimedean complete fields (it includes the case of the finite separable extensions, and also the case of any complete extension of a perfect complete non-archimedean field) and show that the usual commutative algebra properties (Rm, Sm, Gorenstein, Cohen-Macaulay, Complete Intersection) are stable under analytically separable ground field extensions; we also establish a GAGA principle with respect to those properties for any finitely generated scheme over an affinoid algebra. A second part of the paper deals with more global geometric notions : we define, show the existence and establish basic properties of the irreducible components of analytic space ; we define, show the existence and establish basic properties of its normalization ; and we study the behaviour of connectedness and irreducibility with respect to base change.Comment: This is the (almost) definitive version of the paper, which is going to appear in "Annales de l'institut Fourier