Some properties of G-schemes

TL;DR

This paper extends the concepts of spectra and schemes from algebraic geometry to the category of groups, exploring properties like irreducibility, radicals, and sheaves in this new context.

Contribution

It introduces a new notion of prime ideals for G-schemes and studies their properties, advancing the algebraic geometric framework within group theory.

Findings

Defined a new topology on prime ideals of G-schemes

Analyzed properties like irreducibility and radicals in G-spectra

Developed concepts of G-varieties and G-schemes

Abstract

In this paper, we continue to adapt the theories of spectra and schemes developed by Grothendieck in algebraic geometry to the category of groups. Let $G$ be a group, and $(H,f_G^H)$ and object of the comma category $C(G)$. In [5], we have defined on the set $Spec_G(H)$ of prime ideals of $(H,f_G^H)$ a topology. In this paper, we define another notion of prime ideals to which we associate a spectrum endowed with a topology. We study some properties and objects associated to these spectra; amongst them we can quote, irreducibility, the radical, the structural sheaf, $G$-varieties and $G$-schemes. \bigskip