Abstract. In previous work, the second author introduced a topology, for spaces of irreducible representations, that reduces to the classical Zariski topology over commutative rings but provides a proper refinement in various noncommutative settings. In this paper, a concise and elementary description of this refined Zariski topology is presented, under certain hypotheses, for the space of simp...

— 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...

$r$-module. in this paper, we explore more properties of $max$-injective modules and we study some conditions under which the maximal spectrum of $m$ is a $max$-spectral space for its zariski topology.

In the classical case all unirational surfaces are rational. This was rst realized by Oscar Zariski ((20], p. 314). Prompted by Hironaka's suggestion, we began an investigation of the type of surfaces introduced by Zariski in that paper. The research was originally begun by Blass in 1970-71 at Harvard with the advice of Hironaka and Zariski, and then during 1974{1977 he continued under the dire...

We formulate and prove a generalization of Zariski-van Kampen theorem on the topological fundamental groups of smooth complex algebraic varieties. As an application, we prove a hyperplane section theorem of Lefschetz-Zariski-van Kampen type for the fundamental groups of the complements to the Grassmannian dual varieties.

4 Zariski sheaves associated with pretheories. 25 4.1 Technical lemmas. . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 Pretheories in a neighborhood of a smooth subscheme. . . . . . 31 4.3 Pretheories on curves over a field. . . . . . . . . . . . . . . . . 35 4.4 Pretheories on semi-local schemes. . . . . . . . . . . . . . . . . 36 4.5 Zariski cohomology of sheaves associated with pre...