Let $M$ and $N$ be two finitely generated graded modules over a standard graded Noetherian ring $R=bigoplus_{ngeq 0} R_n$. In this paper we show that if $R_{0}$ is semi-local of dimension $leq 2$ then, the set $hbox{Ass}_{R_{0}}Big(H^{i}_{R_{+}}(M,N)_{n}Big)$ is asymptotically stable for $nrightarrow -infty$ in some special cases. Also, we study the torsion-freeness of graded generalized local ...

Let $K$ be a field. The \'etale open topology on the $K$-points $V(K)$ of $K$-variety $V$ was introduced in our previous work. is non-discrete if and only large. If separably, real, $p$-adically closed then agrees with Zariski, order, valuation topology, respectively. We show that existentially definable sets perfect large fields behave well respect to this topology: such are finite unions subs...

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...

Here we take the view that small abelian categories are certain categorical versions of rings and the collection of Serre subcategories of a small abelian category is then analogous to the set of ideals of a commutative ring. We consider two topologies on the collection of Serre subcategories; one is a very direct lift of the Zariski topology, the other, dual, topology appeared in the context o...

The starting point for this dissertation is whether the concept of Zariski geometry, introduced by Hrushovski and Zilber, could be generalized to the context of nonelementary classes. This leads to the axiomatization of Zariski-like structures. As our main result, we prove that if the canonical pregeometry of a Zariski-like structure is non locally modular, then the structure interprets either ...

We relate closure operations for ideals and for submodules to non-flat Grothendieck topologies. We show how a Grothendieck topology on an affine scheme induces a closure operation in a natural way, and how to construct for a given closure operation fulfilling certain properties a Grothendieck topology which induces this operation. In this way we relate the radical to the surjective topology and...

For a module M over commutative ring R with identity, let RSpec(M) denote the collection of all submodules L Msuch that ?(L:M) is prime ideal and equal to (rad L:M). In this article, we topologies topology which enjoys analogs many properties Zariski on spectrum Spec(M) (as subspace topology). We investigate topological space from point view spectral spaces by establishing interrelations betwee...

Our goal for this lecture is to prove that morphisms of projective varieties are closed maps. In fact we will prove something stronger, that projective varieties are complete, a property that plays a role comparable to compactness in topology. For varieties, compactness as a topological space does not mean much because the Zariski topology is so coarse. Indeed, every subset of An (and hence of ...