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Introduction to Non Commutative Algebraic Geometry | OMICS International
ISSN: 2090-0902
Journal of Physical Mathematics
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Introduction to Non Commutative Algebraic Geometry

Arvid Siqveland*

Dean, Faculty of Technology, Buskerud University College Kongsberg, Norway

*Corresponding Author:
Siqveland A
Faculty of Technology
Buskerud University College Kongsberg, Norway
Tel: +4731008918
E-mail: [email protected]

Received Date: February 02, 2015; Accepted Date: April 11, 2015; Published Date: April 21, 2015

Citation: Siqveland A (2015) Introduction to Non Commutative Algebraic Geometry. J Phys Math 6:133. doi:10.4172/2090-0902.1000133

Copyright: © 2015 Siqveland A. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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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

Keywords

Non commutative algebraic geometry; Topology; Transition morphisms

Introduction

Algebraic varieties

In this introduction, we use Hartshornes classical book on algebraic geometry [1,2] as reference. We consider the free polynomial algebra over equation, Char k = 0, A=k [t1… td]. The affine n-space is the set of points in equation, an algebraic set is given by an ideal equation as the zero set equation. . The algebraic sets are the closed sets in a topology on equationcalled the Zariski topology, and an algebraic, affine variety is a closed, irreducible (i.e. it is not a union of two proper closed subsets, equivalently, every open subset is dense), subset of equation . One basic term in algebraic varieties is an arrow-reversing correspondence from closed subsets equation to radical ideals equation . The ideal of a closed subset is equation

Thus equation

There is a close connection between differential geometry and algebraic geometry, and because differential geometry is seen as a tool for applications (physics), the same is true for algebraic geometry. The topology in differential geometry is the smallest topology making the analytic functions continuous. In algebraic geometry, we work with polynomials rather that power-series, so we use the smallest topology that makes rational functions continuous. That is the Zariski topology defined above.

In the Zariski topology, we have the definition of regular functions:

Let equation be an open subset.A ?:u→k is called regular if there exists polynomials f, h such that equation with equation for all P∈U .

Definition 1: The ring of regular functions defined over an open subset equation is the ring equation with its natural ring operations.

Definition 2: (Inductive and Projective Limits). A directed set (I ≤?) is a partially ordered set such I that every finite subset of elements has an upper bound, or equivalently, that for each pair a,b ? I there is a c ? I such that a ≤ c and b ≤ c. Consider a small category a.

a) A projective system of elements in a is a family of objects equation together with transition morphisms equation for each pair equation with the properties that, for each equation, and if i ≤ j ≤ k then equation The projective limit of the projective system is defined as an object equation a with morphisms equation for each i such that for all equation , and such that if equation is another object with corresponding properties, then there is a unique morphism equation such that equation. In a small category, to prove the unique existence of projective limits, we let

equation

b) An Inductive system is the dual of a projective: It is a family of objects equation together with transition morphisms equation for each pair equation with the properties that, for each equation and if i ≤ j ≤ k then equation

The inductive limit of the inductive system is defined as an object equation with morphisms equation for each i such that for all i≤j equation and such that if equation is another object with corresponding properties, then there is a unique morphism equation such that equation. In a small category, to prove the unique existence of inductive limits, we let

equation

For the definitions in this text, we notice that the family of open subsets is a directed set partially ordered by inclusion. The ring of regular functions locally at P is equation , and by duality, we also have the other way around: equation

By this we have that the coordinate ring of the variety V is equation

And that the ring of locally regular functions in P is

equation

Where equation is the maximal ideal corresponding to P

The final definition of the category of affine varieties in the commutative situation is the definition of morphisms. Morphism between two affine varieties V,W is a continuous map equation such that the induced map equation is well defined for each open equation , that is equation is regular on V.

Local Categories

Everything in this section and the next can be found in M. Schlessinger's classical work [7]. Let equation denote the category of local artinian k-algebras with residue field k that is diagrams

equation

with A local, artinian. The morphisms in l are the k-algebra homo morphisms. We let equation enote the pro category, which is the category of projective limits in l. For any covariant functor f?c→ Sets we have the following lemma:

Lemma 1 (Yoneda): For any object C ? C there is an isomorphism

equation

Given by equation, with inverse equation

The lemma extends to procategories, and is true for contra variant functors when we replace equationwith mor(-,C) . In particular:

Lemma 2: Let f?l→ Sets be a covariant functor. Then for everyequation there is an isomorphism

equation

As usual equation denotes the algebra of dual numbers. An epi morphism π: R→S in l is called small if equation where mR is the maximal ideal In R . Finally, a transformation of functors f,g?l→Sets is smooth if for any small morphismsequation , in the diagram

equation

if objects equation and equation there is an object equation mapping to both equation and equation

The following concept is the one we generalize in this text:

Definition 3: The couple equation is said to pro represent F if equation is an isomorphism. The couple is said to be a pro representing hull, or equation is said to be a formal moduli with proversal family, equationif equation is smooth and an isomorphism for equation,(usually and reasonably) called the tangent level.

Lemma 3: A pro representing object is unique up to unique isomorphism. A pro representing hull is unique up to non unique isomorphism.

Global to Local Theory

Let f?sch?k→s Sets be a covariant functor. Assume there exists a fine moduli space for the set F (k) (which can be interpreted by the "family"-functor being representable). This means that there exists a scheme M/k and a universal family ??F(M) such that, with the notation above, equationis an isomorphism. Let M ? F (Spec k) be an object represented by the closed point equation, and de ne a covariant functor equation

equation

Because M is a fine moduli, equation.

This says that equation prorepresent equation and so is unique up to unique isomorphism.

We call equation the local deformation functor. The idea is the following:

The local formal moduli represent the local, completed rings of the moduli scheme, and can be used to analyse, or to construct, the moduli scheme.

Algebraic Varieties Revisited (Defined by local theory)

Let equationbe a k-algebra. Then Spec A is fine moduli for its closed points (maximal ideals). A point m∈ A Spec A corresponds to a unique morphismequation, i.e.

equation

Definition 4: Let M be an A-module. Then equationSets is defined by

equation

where two deformations are equivalent if there is an isomorphism equation commuting with the fibre, i.e.

equation

The earlier discussion shows that if equation for equation maximal, then equation pro represents equation. Thus the affine theory can be defined as before, but with the local rings replaced by local formal moduli in each point. This is, by the way, the way we use deformation theory to construct moduli.

Notice that we have an injection equation by definition, because an equation -structure on equation , at over S, is a homo morphism equation.

Non Commutative Affine Algebraic Geometry

For the ordinary, commutative affine algebraic geometry, the basic object is the polynomial algebra in d ? N variables. In the non commutative situation, we take the matrix polynomial algebra as our basic object. That is:

Let equation be an equation -matrix. Then the matrix polynomial algebra equation is the r × r matrix polynomial algebra generated by the idem potents equation together with the matrix variables equation for 1 ≤ i, j ≤ r. We use the notation

equation

Notice that we use the commutative polynomial k -algebras on the diagonal. This is not the natural free object in the category, but we use it because it is simpler to give a (naive) geometric interpretation. Also notice that this notation implies that the multiplication of equationare given by matrix multiplication.

Example 1: equation

Differential geometry was generalized to noncommutative geometry by Connes and Marcolli [3], and further developed by Dubois-Violette in [4].

Generalization of differential geometry to matrix algebras is given by Dubois-Violette, Kerner and Madore in [5].

We use results from the above referred articles in the generalization of algebraic geometry. Before we are ready to define the noncommuative analogue of the ring of dual numbers:

Definition 5: The non commutative r × r k-algebra of dual numbers, also called the test algebra, is the algebra equation

where D is the r×r-matrix with 1 in every entry, and m is the ideal generated by all the variables equation

The rest of the results in this section can be found in the work of Arnfinn Laudal [6].

Definition 6: The category ar is the category with objects Artinian algebras fitting in the diagram

equation

and such that equation for some equationand with morphisms the- Kr-algebra homo morphisms commuting with the above diagrams.

Definition 7: Let equationbe a set of equationright A -modules, and put equationThen we define equation

equation

the relation equationbeing the one corresponding to the commutative situation. We must assume equationto be s ? a abi module on which k acts centrally.

Notice that the propertyequationsays that the isomorphism is as S-modules. This is equivalent to Ms being S-flat, but we take that into the definition.

Definition 8: HM is called semilocal formal moduli with formally versal family M if equation is smooth, and an isomorphism for the testalgebra.

Lemma 4: The non commutative deformation functor equation has a semi local moduli determined by some welldefined Generalized Massey Products. That is to say, it can be constructed. Also, the construction gives a well-defined injection [7].

equation

Proof: See Eriksen [1] or Siqveland [8] for a constructive proof. A proof of existence can be given by generalizing the classical proof of Schlessinger in [7] verbatim.

Now we have all the needed tools necessary to define the noncommuative affine space.

Definition 9: Consider a matric polynomial algebra equation, equation . The affine algebraic space equation of this algebra is the disjoint union of the affine spaces on the diagonal, that is equation with the product (Zariski) topology. Each (closed) point in this space corresponds to a maximal ideal on the diagonal in the matrix algebra, which again corresponds to one-dimensional representations of A. For each finite set of (closed) points equation , we let equation and we define the semi local ring of equation in V as equation

The generalized concept of localization immediately gives the natural generalizations of affine varieties, regular maps, and morphisms. A lot of result needs to be established, which we will do in forthcoming work. Also, the deformation theory can be removed from the discussion, by defining the semi-local rings by their generalized Massey Products which can be given intrinsic.

Also, as algebraic geometry can be seen as a simplification of differential geometry for physical models, the noncommutative theory is needed for physical models involving entanglement.

For more examples, see the author’s articles [9-11] where more examples appear as resulting algebras of noncommutative deformation theory.

Acknowledgements

I would like to thank Daniel Larsson and Olav Imenes for the help and discussion in the development of this naive theory. Of course, nothing could have been written without Eivind Eriksens thorough cooperation, and Arnfinn Lau-dal's main conducting. Also, I would like to thank the anonymous referees for corrections and suggestions improving the article.

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