Triangulations et cohomologie étale sur une courbe analytique $p$-adique

Let $k$ be a non-Archimedean complete field and let $X$ be a $k$-analytic curve as defined by Berkovich. Let $\pi$ be the natural map between the étale site and the underlying topological space of $X$. This text is devoted to the study of the cohomology of sheaves of the kind $\mbox {R}^{q}\pi _{*}\mathscr F$. First of all, we define what a triangulation of a curve is and we prove, using the semi-stable reduction theorem, that any curve whose singular locus is nowhere dense has a triangulation. For given $q$ and $\mathscr F$ we associate to any triangulation on $X$ a two-term complex whose cohomology groups are shown to be precisely those of $\mbox {R}^{q} \pi _{*}\mathscr F$.

This allows us to prove a comparison theorem between our groups and their scheme-theoretic counterparts when $X$ is the analytification of a smooth algebraic $k$-curve. After that we assume that $k$ comes with a dualizing module (e.g., $k={\mathbb C} ((t))$ or $k={\mathbb Q}_{p}).$ Then we build some pairings between suitable cohomology groups and show that their non-degeneratedness is equivalent to some arithmetic properties of some one-dimensional function fields over $\widetilde {k}$ which are attached to certain points of $X;$ thanks to this equivalence we can prove that we actually have perfect pairings in some cases.

Putting those results all together we recover as corollaries some previous theorems of the author concerning unramified $\mbox {H}^{3}$ with coefficients $\mu _{n}^{\otimes 2}$ over a $p$-adic curve, and also Lichtenbaum’s duality between the Picard group and the Brauer group of such a curve.