NORMAL AND IRREDUCIBLE ADIC SPACES, THE OPENNESS OF FINITE MORPHISMS, AND A STEIN FACTORIZATION

Abstract

AbstractWe transfer several elementary geometric properties of rigid-analytic spaces to the world of adic spaces, more precisely to the category of adic spaces which are locally of (weakly) finite type over a non-archimedean field. This includes normality, irreducibility (in particular, irreducible components), and a Stein factorization theorem. Most notably, we show that a finite morphism in our category of adic spaces is automatically open if the target is normal and both source and target are of the same pure dimension. Moreover, our version of the Stein factorization theorem includes a statement about the geometric connectedness of fibers which we have not found in the literature of rigid-analytic or Berkovich spaces.