Introduction to Non Commutative Algebraic Geometry

Introduction Algebraic varieties In this introduction, we use Hartshornes classical book on algebraic geometry [2] as reference. We consider the free polynomial algebra over = k k , Char k = 0, A=k [t1... td]. The affine n-space is the set of points in =  n k , an algebraic set is given by an ideal ⊆ a A as the zero set ( ) { | ( ) 0 } = ∈ = ∀ ∈  a a n Z P f P f . The algebraic sets are the closed sets in a topology on  called 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 ⊆  V . One basic term in algebraic varieties is an arrow-reversing correspondence from closed subsets ( ) ⊆  a n Z to radical ideals ⊆ a A . The ideal of a closed subset is ( ) { | ( ) 0 } = ∈ = ∀ ∈ I V f A f P P V


Introduction Algebraic varieties
In this introduction, we use Hartshornes classical book on algebraic geometry [2] as reference. We consider the free polynomial algebra over = k k , Char k = 0, A=k [t 1 … t d ]. The affine n-space is the set of points in =  n n k , an algebraic set is given by an ideal ⊆ a A as the zero set . The algebraic sets are the closed sets in a topology on  n called 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 ⊆  n V . One basic term in algebraic varieties is an arrow-reversing correspondence from closed subsets ( ) ⊆  a n Z to radical ideals 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 a small category, to prove the unique existence of projective limits, we let An Inductive system is the dual of a projective: It is a family of The inductive limit of the inductive system is defined as an object lim  By this we have that the coordinate ring of the variety V is And that the ring of locally regular functions in P is 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 : φ

Local Categories
Everything in this section and the next can be found in M. Schlessinger's classical work [7]. Let  denote the category of local artinian k-algebras with residue field k. That is diagrams with A local, artinian. The morphisms in  are the k-algebra homo morphisms commiting in the diagram. We let  denote the procategory, which is the category of projective limits in l. For any covariant functor f꞉C→ Sets we have the following lemma: The lemma extends to procategories, and is true for contra variant functors when we replace mor(C,-) with mor(-,C) . In particular: The following concept is the one we generalize in this text: is an isomorphism. The couple is said to be a prorepresenting hull, or R is said to be a formal moduli with proversal family, ˆ, ξ if ˆ( ) ψ ξ R is smooth and an isomorphism for [ ] ε k , (usually and reasonably) called the tangent level.

Lemma 3:
A prorepresenting object is unique up to unique isomorphism. A prorepresenting 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 uЄF(M) such that, with the notation above, 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.

Non Commutative Affine Algebraic Geometry
For the ordinary, commutative affine algebraic geometry, the basic object is the polynomial algebra in d Є N variables. In the non commutative situation, we take the matrix polynomial algebra as our basic object. That is: The affine algebraic space  D of this algebra is the disjoint union of the affine spaces on the diagonal, that is 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 ) 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][10][11] where more examples appear as resulting algebras of noncommutative deformation theory.