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...

A meadow is a zero totalised field (0 = 0), and a cancellation meadow is a meadow without proper zero divisors. In this paper we consider differential meadows, i.e., meadows equipped with differentiation operators. We give an equational axiomatization of these operators and thus obtain a finite basis for differential cancellation meadows. Using the Zariski topology we prove the existence of a d...

We undertake a case study of two series of nonclassical Zariski geometries. We show that these geometries can be realised as representations of certain noncommutative C∗-algebras and introduce a natural limit construction which for each of the two series produces a classical U(1)-gauge field over a 2-dimensional Riemann surface.

We show that if a group automorphism of Cremona arbitrary rank is also homeomorphism with respect to either the Zariski or Euclidean topology, then it inner up field base-field. Moreover, we similar result holds consider groups polynomial automorphisms affine spaces instead groups.

We describe the structure QHO = QHON (dependent on the positive integer number N) on the universe L which is a finite cover, of order N, of the projective line P = P(F), F an algebraically closed field of characteristic 0. We prove that QHO is a complete irreducible Zariski geometry of dimension 1. We also prove that QHO is not classical in the sense that the structure is not interpretable in a...

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.