An Introduction to Zariski Spaces over Zariski Topologies

Zariski Dense Random Walks on Homogeneous Spaces

ZARISKI-LIKE SPACES OF CERTAIN MODULES

Let $R$ be a commutative ring with identity and $M$ be a unitary$R$-module. The primary-like spectrum $Spec_L(M)$ is thecollection of all primary-like submodules $Q$ such that $M/Q$ is aprimeful $R$-module. Here, $M$ is defined to be RSP if $rad(Q)$ isa prime submodule for all $Qin Spec_L(M)$. This class containsthe family of multiplication modules properly. The purpose of thispaper is to intro...

Fiber Bundles in Analytic, Zariski, and Étale Topologies

We compare the behavior of fiber bundles with structure group G for G the general linear group GL, the projective general linear group PGL, and the general affine group GA. We prove a criterion for when a GA-bundle is in fact a GL-bundle, i.e., a vector bundle, over the Riemann sphereP1. We also discuss the behavior of fiber bundles under different topologies; specifically, we compare the analy...

Relative Riemann - Zariski

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...

An Algebraic Approach to Zariski Problem

Zariski Problem (Cancellation of indeterminates) is settled affirmatively, that is, it is proved that : Let k be an algebraically closed field of characteristic zero and let n, m ∈ N. If R[Y1, . . . , Ym] ∼=k k[X1, . . . , Xn+m] as k-algebras, where Y1, . . . , Ym, X1, . . . , Xn+m are indetermoinates, then R ∼=k k[X1, . . . , Xn]. Zariski Problem is the following : Zariski Problem. Let k be an...