We investigate the Zariski Topology on the L-prime spectrum of modules consisting of the collection of all prime Lsubmodules and prove some useful results.

The characteristic polynomials of geometric automorphisms of a free group of finite rank at least three form a nowhere dense set in the Zariski topology.

variety). For each i, let φi(x) be a C-dense quantifier-free Li-formula with parameters from K. Then we can find a K-definable rational function f : C → P which is non-constant, and has the property that the divisor f−1(0) is a sum of distinct points in ⋂n i=1 φi(K), with no multipliticities. (In particular, the support of the divisor contains no points from C(K)\C(K) and no points from C \ C.)...

Classical algebraic geometers studied algebraic varieties over the complex numbers. In this setting, they didn’t have to worry about the Zariski topology and its many pathologies, because they already had a better-behaved topology to work with: the analytic topology inherited from the usual topology on the complex numbers themselves. In this note, we introduce the analytic topology, and explore...

Let R be a commutative ring with nonzero identity and M an R-module. In this paper, first we give some relations between S-prime S-maximal submodules that are generalizations of prime maximal submodules, respectively. Then construct topology on the set all , which is generalization spectrum M. We investigate when SpecS(M) T0 T1-space. also study continuous maps irreducibility SpecS(M). Moreover...

In this paper, we introduce and study a new topology related to a self mapping on a nonempty set.Let X be a nonempty set and let f be a self mapping on X. Then the set of all invariant subsets ofX related to f, i.e. f := fA X : f(A) Ag P(X) is a topology on X. Among other things,we nd the smallest open sets contains a point x 2 X. Moreover, we find the relations between fand To f . For insta...

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...

In general, the sheaf criterion on the étale topology may be difficult to verify directly, as a scheme will in general have many étale covers. It is clear that a necessary condition for a presheaf F to be a sheaf on Xet is that it be a sheaf with respect to Zariski covers (i.e., its restriction to Xzar is a sheaf), and that it be a sheaf with respect to one-piece étale covers (V → U) such that ...