Abstract The aim of this paper is to investigate the behaviour prime and semiprime subgroups groups, their relation with existence abelian normal subgroups. In particular, we study set Spec( G ) all a group endowed Zariski topology and, among other examples, construct an infinite whose proper are form descending chain type ? + 1.

Let $$C \rightarrow \mathop {{\mathrm{Spec}}}\nolimits (R)$$ be a relative proper flat curve over henselian base. G reductive C-group scheme. Under mild technical assumptions, we show that G-torsor C which is trivial on the closed fiber of locally for Zariski topology.

We factor the virtual Poincaré polynomial of every homogeneous space G/H , where G is a complex connected linear algebraic group and H is an algebraic subgroup, as t(t − 1)QG/H(t ) for a polynomial QG/H with non-negative integer coefficients. Moreover, we show that QG/H(t ) divides the virtual Poincaré polynomial of every regular embedding of G/H , if H is connected. Introduction and statement ...

It appears that if dimcZ = a. dim X > 2, one may consider X as an algebraic variety too but in some new sense. One can generalize the conception of the abstract variety of A. Weil by substituting the etale topology of Grothendieck for the topology of Zariski. One gets the objects which M. Artin called "etale schemes" and the author called "minischemes". Later, M. Artin introduced the term "alge...

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...

For a compact HyperKähler manifold X , we show certain Zariski decomposition for every pseudo-effective R-divisor, and give a sufficient condition for X to be bimeromorphic to a Lagrangian fibration.

The main purpose of this paper is to provide a survey of different notions of algebraic geometry, which one may associate to an arbitrary noncommutative ring R. In the first part, we will mainly deal with the prime spectrum of R, endowed both with the Zariski topology and the stable topology. In the second part we focus on quantum groups and, in particular, on schematic algebras and show how a ...

in this paper‎, ‎we mainly investigate how the generalized metrizability properties of the remainders affect the metrizability of rectifiable spaces‎, ‎and how the character of the remainders affects the character‎ ‎and the size of a rectifiable space‎. ‎some results in [a. v‎. ‎arhangel'skii and j‎. ‎van mill‎, ‎on topological groups with a first-countable remainder‎, ‎topology proc. 42 (2013...

Let X be a variety defined over an algebraically closed field k of characteristic 0, let N ∈ N, let g : X99KX be a dominant rational self-map, and let A : AN−→AN be a linear transformation defined over k(X), i.e., for a Zariski open dense subset U ⊂ X, we have that for x ∈ U(k), the specialization A(x) is an N -by-N matrix with entries in k. We let f : X × AN99KX × AN be the rational endomorphi...