Let R be an associative ring with identity. We study an elementary generalization of the classical Zariski topology, applied to the set of isomorphism classes of simple left Rmodules (or, more generally, simple objects in a complete abelian category). Under this topology the points are closed, and when R is left noetherian the corresponding topological space is noetherian. If R is commutative (...

The main result implies that a proper convex subset of an irreducible higher rank symmetric space cannot have Zariski dense stabilizer.

the notions of quasi-prime submodules and developed  zariski topology was introduced by the present authors in cite{ah10}. in this paper we use these notions to define a scheme. for an $r$-module $m$, let $x:={qin qspec(m) mid (q:_r m)inspec(r)}$. it is proved that $(x, mathcal{o}_x)$ is a locally ringed space. we study the morphism of locally ringed spaces induced by $r$-homomorphism $mrightar...

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...

We ask whether the notion of a homotopy class of a path on a complex algebraic variety admits a purely algebraic characterisation, and reformulate this question as a question of categoricity of the universal covering space of a complex algebraic variety in a natural countable in nitary language. We provide partial positive results towards the question. Assuming a conjecture of Shafarevich and s...

In this short note, we study the geometry of the eigenvariety parametrising p-adic automorphic forms for GL1 over a number field, as constructed by Buzzard. We show that if K is not totally real and contains no CM subfield, points in this space arising from classical automorphic forms (i.e. algebraic Grössencharacters of K) are not Zariski-dense in the eigenvariety (as a rigid space); but the e...

1 Affine schemes 4 1.1 Motivation and review of varieties . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 First attempt at defining an affine scheme . . . . . . . . . . . . . . . . . . . 4 1.3 Affine schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 The Zariski topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 The ringed space s...

We present a new probabilistic symbolic algorithm that, given a variety defined in an n-dimensional affine space by a generic sparse system with fixed supports, computes the Zariski closure of its projection to an l-dimensional coordinate affine space with l < n. The complexity of the algorithm depends polynomially on combinatorial invariants associated to the supports.

The notions of quasi-prime submodules and developed  Zariski topology was introduced by the present authors in cite{ah10}. In this paper we use these notions to define a scheme. For an $R$-module $M$, let $X:={Qin qSpec(M) mid (Q:_R M)inSpec(R)}$. It is proved that $(X, mathcal{O}_X)$ is a locally ringed space. We study the morphism of locally ringed spaces induced by $R$-homomorphism $Mrightar...