Introduction to Non Commutative Algebraic Geometry

Arvid Siqveland

doi:10.4172/2090-0902.1000133

Abstract

Ordinary commutative algebraic geometry is based on commutative polynomial algebras over an algebraically closed eld k. Here we make a natural generalization to matrix polynomial k-algebras which are non-commutative coordinate rings of non-commutative varieties

Highlights

In the Zariski topology, we have the definition of regular functions: Let U ⊆ n be an open subset
Commutative affine algebraic geometry, the basic object is the polynomial algebra in d Є N variables
The generalized concept of localization immediately gives the natural generalizations of affine varieties, regular maps, and morphisms

Summary

Introduction

In the Zariski topology, we have the definition of regular functions: Let U ⊆ n be an open subset. B) An Inductive system is the dual of a projective: It is a family of objects {Ai}i∈I together with transition morphisms ψ ij : Ai → Aj for each pair i ≤ j ∈ I with the properties that, for each i ∈ I ,ψ ii = id and if i ≤ j ≤ k ψ jk °ψ ij = ψ ik . Morphism between two affine varieties V,W is a continuous map φ :V → W such that the induced map φ # : W (U ) → V (φ −1(U )) is well defined for each open U ⊆ W , that is f f °φ is regular on V.

Results

Conclusion