0965 Contents 1. Introduction 1 2. Some topology 2 3. Local isomorphisms 4 4. Ind-Zariski algebra 6 5. Constructing w-local affine schemes 6 6. Identifying local rings versus ind-Zariski 10 7. Ind-´ etale algebra 14 8. Constructing ind-´ etale algebras 15 9. Weaklyétale versus pro-´ etale 18 10. Constructing w-contractible covers 18 11. The pro-´ etale site 21 12. Points of the pro-´ etale site...

Let k be an algebraically closed field andK be a finitely generated k-field. In the first half of the 20-th century, Zariski defined a Riemann variety RZK(k) associated to K as the projective limit of all projective k-models of K. Zariski showed that this topological space, which is now called a Riemann-Zariski (or Zariski-Riemann) space, possesses the following set-theoretic description: to gi...

Based on a recent work of Thomas Bauer’s [1] reproving the existence of Zariski decompositions for surfaces, we construct a b-divisorial analogue of Zariski decomposition in all dimensions.

A complex affine Poisson algebra is said to satisfy the Dixmier-Moeglin equivalence if cores of maximal ideals are precisely those prime that locally closed in spectrum P.spec and if, moreover, these whose extended centers exactly numbers. In this paper, we provide some topological criteria for terms poset (P.spec A, ⊆) symplectic leaf or core stratification on its spectrum. particular, prove Z...

is an isomorphism. Here F : X −→ X denotes the Frobenius morphism on X and H denotes the a cohomology sheaf of F∗Ω•X . If the variety is not smooth, not much is known about the properties of the Cartier operator and the poor behaviour of the deRham complex in this case makes its study difficult. If one substitutes the deRham complex with the Zariski-deRham complex the situation is better. For e...

We begin by showing that commensurators of Zariski dense subgroups of isometry groups of symmetric spaces of non-compact type are discrete provided that the limit set on the Furstenberg boundary is not invariant under the action of a (virtual) simple factor. In particular for rank one or simple Lie groups, Zariski dense subgroups with non-empty domain of discontinuity have discrete commensurato...

1.1. Sites. We begin with the underlying idea for a Grothendieck topology. Consider your favorite topological space X. Its topology, don’t use the Zariski topology, is determined by the partially ordered category UX consisting of all the open subsets of X ordered with respect to inclusion. In order to verify that a presheaf is a sheaf on X, we must first construct all possible coverings U = (Ui...

— In this paper we show how to recover a special class of valuations (which generalize in a natural way the Zariski prime divisors) of function fields from the Galois theory of the functions fields in discussion. These valuations play a central role in the birational anabelian geometry and related questions. Résumé (Théorie de Galois pro-` des diviseurs premiers de Zariski) Dans cet article nou...

We give a cohomological characterization of semiample line bundles. The result is a common generalization of the Fujita-Zariski Theorem on semiampleness and the Grothendieck-Serre criterion for ampleness. As an application of the Fujita-Zariski Theorem we characterize contractible curves in 1-dimensional families.

Let G be a semi-simple Lie group and Γ < G be a lattice. This paper is motivated by the attempt to understand the infinite index subgroup structure of Γ . In particular, to understand the possibilities for infinite index, finitely generated, freely indecomposable, Zariski dense subgroups of Γ . The study of Zariski dense subgroups of semi-simple Lie groups has a long and rich history. Some high...