Geometry of Noncommutative k-Algebras

Let X be a scheme over an algebraically closed field k, and let x ∈ SpecR ⊆ X be a closed point corresponding to the maximal ideal m ⊆ R. Then ÔX,x is isomorphic to the prorepresenting hull, or local formal moduli, of the deformation functor DefR/m : → Sets. This suffices to reconstruct X up to etalé coverings. For a noncommutative k-algebra A the simple modules are not necessarily of dimension one, and there is a geometry between them. We replace the points in the commutative situation with finite families of points in the noncommutative situation, and replace the geometry of points with the geometry of sets of points given by noncommutative deformation theory. We apply the theory to the noncommutative moduli of three-dimensional endomorphisms. MSC 2010: 14A22, 14D22, 14D23, 16L30


Introduction
There have been several attempts to generalize the ordinary commutative algebraic geometry to the noncommutative situation. The main problem in the direct generalization is the lack of localization of noncommutative k-algebras. This can only be done for Ore sets, and does not give a satisfactory solution to the problem.
In the study of flat deformations of A-modules when A is a commutative, finitely generated k-algebra (k algebraically closed), one realizes that for each maximal ideal m, putting V = A/m, the deformation functor Def V has a (unique up to nonunique isomorphism) prorepresenting hull (local formal moduli)Ĥ(V ) isomorphic to the completed local ring, that isĤ(V ) ∼ =Âm, see [5].
In the general situation with A not necessarily commutative, the deformation theory can be directly generalized to families of right (or left) A-modules, see [1] or [3], and we can replace the local complete rings with the local formal moduli of finite subsets of the simple modules. From now on, k denotes an algebraically closed field of characteristic zero. An A-module M is simple if it contains no other proper submodules but the zero module (0); it is indecomposable if it is not the sum of two proper submodules.
The following results from Eriksen [1] and Laudal [3] are assumed as a basis for this text.
Definition 1. ar is the category of r-pointed Artinian k-algebras. An object of this category is an Artinian k-algebra R, together with a pair of structural ring homomorphisms f : k r → R and g : R → k r with g • f = Id, such that the radical I(R) = ker(g) is nilpotent. The morphisms of ar are the ring homomorphims that commute with the structural morphisms.
For any family V = {V 1 , . . . , Vr} of right A-modules, there is a noncommutative deformation functor Def V : Laudal (or equally Eriksen) proves that Def V has a formal moduli (Ĥ,MĤ ), unique up to nonunique isomorphism. Given this, the local reconstruction theorem is the following. Theorem 2 (the generalized Burnside theorem). Let A be a finite dimensional k-algebra, and let V = {V 1 , . . . , Vr} be the family of simple right A-modules. Then, the (Ĥ-flat) proversal family η : We will use Laudal and Eriksen's results to define (geometric) formal localizations, and use this to define the noncommutative affine spectrum Spec A. This leads to the definition of a noncommutative variety and its relation to noncommutative moduli. We will end the paper with a classical example, the moduli of 3 × 3-matrices up to conjugacy.
This article is a part of a Special Issue on Deformation Theory and Applications (A. Makhlouf, E. Paal and A. Stolin, Eds.).

r-pointed ringed spaces
Lemma 3. Let A be a finitely generated, commutative k-algebra and m 1 , m 2 two different maximal ideals with corresponding simple Proof. It is enough to consider The inner derivations are given by adγ (x i ) = γx i − x i γ = γα i in this case, and this determines the (inner) derivations completely. Now, let δ : A → Hom(V 1 , V 2 ) be a derivation. Then, since A is commutative, which proves that every derivation is inner.
In the noncommutative case, the above result is obviously no longer true, so that if a scheme should be a classifying space for the simple modules of a noncommutative k-algebra, it should consider sets of points and their infinitesimal geometry. This is then necessary for the reconstruction of k-algebras in general. We will see that in some cases this is also sufficient.

Matrix algebras
To ease the explicit understanding of noncommutative varieties, we now treat the explicit case here. To introduce notation, we give an example with an obvious generalization. The free 2 × 2 matrix k-algebra generated by these elements by ordinary matrix multiplication is then denoted We consider the two-sided ideal in F generated by f 11 , that is a = f 11 , and for the quotient algebra we use the notation In this case Q = (Q ij ), and k t 11 (1), t 11 (2) maps injective into Q, but Q 11 = k t 11 (1), t 11 (2) as for example t 12 t 21 ∈ Q 11 . However, letting Q − Q ii be the ideal generated by the matrices in Q with 0 (i, i)-entry, we will write Q 11 = k t 11 (1), t 11 (2) = Q/ Q − Q 11 when necessary.
Let k r → R = (R ij ) be a matrix algebra. We let R − R ii denote the ideal generated by the matrices in R with 0 (i, i)-entry, and we let R ii denote the quotient R/ R − R ii . We call the algebras R ii the diagonal algebras Journal of Generalized Lie Theory and Applications 3 of the matrix algebra R = (R ij ). We let ι ii : R → R/ R − R ii = R ii be the canonical morphism, and we let τ ii : R ii → R be the natural inclusion. Then, τ ii obeys the rules for an algebra morphism except for the fact that τ ii (1) = 1. Thus, τ −1 ii (a) of an ideal a is an ideal.

Proposition 5.
There is a one to one correspondence between the right (left) maximal ideals in the matrix algebra R and the right (left) maximal ideals in its diagonal algebras.
Proof. Let m ⊂ R be a maximal ideal. Then, for some i, is a maximal ideal and together with the canonical surjection ι the correspondence is established.

Geometric localizations
The universal property of the localization L of a commutative k-algebra A in a maximal ideal m is a diagram This definition may very well be extended to the noncommutative situation, but it is well known that the localization process works only for Ore sets. In the following, A is a not necessarily commutative k-algebra.
Proof. Let W be a submodule of V , let 0 = w ∈ W be an element, and let v ∈ V be any element. Let φ : V → V be the linear transformation sending w to v and all other elements in a basis for W to 0. Then, φ = ρa for some a ∈ A because of the surjectivity. Then, v = φ(w) = a · w ∈ W . This proves that V = W and V is simple. The proof of the converse can be found in the introductory book of Lam [2].

Definition 7. Let A be a (not necessarily commutative) k-algebra, and let
and if for any other L with this property, there exists a unique φ : Example 8. As an elementary example, let A be commutative and let m 1 , . . . , mn be maximal ideals.
Notice that the set of simple modules of L are the modules V .
Example 9. Let A be any k-algebra and V 1 , . . . , Vn simple right A-modules. Assume that there exists a k-algebra Knowing that the local formal moduli exists, we can replace the localizations with this. However, we do not know for certain that algebraizations exist. The (next) best we can do is the following: relaxing to some degree the universal property.
Definition 10. Let A be any k-algebra and V = {V 1 , . . . , Vn} a family of simple right A-modules. Then, L is called One writes L =Â V and notices that prolocalizations are not unique.
Then, by the generalized Burnsides theorem, Theorem 2, we have the matrix algebra ( Taking the projective limit, we then end at is a local ring and the general result follows from Proposition 5. Now we come to the main point of this section. For moduli situations, we have to be concerned with the geometry between the different simple objects. This also strengthen the universal property of the localizations we consider. Definition 13 (geometric prolocalizations). Let A be any k-algebra and V = {V 1 , . . . , Vn} a family of simple right A-modules. Then, L is called a geometric prolocalization of A in V if there exists diagrams is a unit for each i, and if there exists an isomorphism of matrix k-algebras We write L =Â G V , and notice that geometric prolocalizations are not unique.

Lemma 14. The geometric prolocalizationÂ
). Then exactly as above,Â G V fulfils the conditions. Notice that even for a noncommutative k-algebra, (u + f )(p − pf p + pf pf p − pf pf pf p + · · · ) = 1 when f ∈ rad(Â G V ) and p is a right unit of u (we recall that rad(Â G V ) = ker η, where η :Â G V → k n is the natural morphism).
If a (geometric) prolocation is finitely generated, we will call it an algebraic localization. This then includes the ordinary localizations.

Noncommutative schemes
For any set S we consider the subset of the power set consisting of finite subsets. We use the notation P (S) = {M ⊆ S | M is finite}. We now make the direct generalization of the sheafification to the noncommutative situation: Journal of Generalized Lie Theory and Applications 5 let A be a not necessarily commutative k-algebra, and put X = Simp(A) = {A-modules V | V is simple}. The generalization of the topological space of A is the Jacobson topology: For f ∈ A, we definê We then define the sheaf of regular, not necessarily commutative, functions on X = Simp A by Now if all theÂ G c are algebraizable, that is, there exist algebraic localizations A G c of A for every finite subset c with natural and coherent morphisms A G c 1 → A G c 2 for each inclusion c 2 ⊆ c 1 , we use the same definition and constructions as above (without the hat) and we end up with the following proposition.
Proposition 15. One has the following:

Proof. (1) We see that A ∼ = A 1 and so this follows by definition. (2) This follows as
Definition 16. We call (Simp A,Ô Simp A ) an affine scheme, and we say that the set of simple A-modules | Simp A| is a scheme for A. A not necessarily commutative scheme is an r-pointed topological space that can be covered by affine schemes.

Relation to moduli problems
Consider any diagram c of A-modules, not necessarily finite. On the set |c|, we define the Jacobson topology generated by the open subsets Dc(f ) for f ∈ A given by Dc is the structure morphism and where End k (V ) * ⊆ End k (V ) denotes the units in this kalgebra. We letÔ V = (Ĥ(i, j) ⊗ k Hom k (V i , V j )) when V = {V 1 , . . . , Vn}. Then, we define a sheaf of r-pointed k-algebras on the topological space |c| as follows. At first, let P (U ) = {c 0 ⊆ c : | c 0 | is finite, c 0 ⊆ U }. Then, we definê This follows directly from the definition.
Given this, we now defineÔ Then,Ôc is a sheaf by the universal property of projective limits which exists in the category of not necessarily commutative k-algebras.

Proposition 17. One has
Proof. When c is finite,Ôc ⊆ (Ĥ ij ⊗ Hom k (V i , V j )) End(V ). As the O-construction is a closure operation and the surjectivity gives simplicity of the representations, dividing out by powers of the radical, using the general Burnside theorem and taking projective limits, the result follows.
Thus, (| c |,Ô | c | ) is a (not necessarily commutative) scheme. Moreover, the natural morphism ρ : A →Ô f glues together to a global moduleρ on | c |. By the geometric properties, it is reasonable to call (| c |,ρ) a moduli for | c |, the original set of A-modules.
Definition 18. c is called an affine scheme for the k-algebra A if (c, Oc) ∼ = (Simp(A), O Simp(A) ).

The noncommutative moduli of rank 3 endomorphisms
In this section, we consider the problem of providing a natural algebraic geometric structure on the set of n×n Jordan forms. It turns out that there are serious combinatorial difficulties in the general case, and also that the general case would be hard to conclude from, in particular geometrically. The case of 2 × 2 Jordan forms can be found in [4], but this example is too simple to illustrate the geometry, thus we restrict to the case of 3 × 3 Jordan forms. The main result of this section is the following.
Theorem 19. The noncommutative k-algebra where b is the two-sided ideal generated by the relations in the generic case (see below), is the algebraic kalgebra of the affine moduli of the GL 3 (k)-orbits of M 3 (k). Thus, it also comes with a universal family, giving the parametrization of the closures of the orbits.
The construction of this structure is based on the noncommutative deformation theory given in [1,3]. Put It will turn out that we have three different cases to consider: the closure of the orbits of the Jordan form with all eigenvalues equal (called the generic case in Theorem 19), the closure of the orbits of the Jordan form with only two different eigenvalues and the closure of the orbit of the Jordan form with three different eigenvalues.

(1) All eigenvalues equal
We are considering the Jordan forms (2) Two different eigenvalues The following Jordan forms are possible, letting λ = (λ 1 , λ 2 ): The case The one and only orbit is the orbit of For simplicity, we will also use the notation We then have the following representation of the ideals defining the closures.

Proposition 22. The k-dimension of Ext 1
A−G (V i , V j ) is given as the (i, j) entry in the matrix ⎛ This is true in all three cases, even if the representation of the orbits differs in notation.
Proof. This is more or less straight forward computations, except for two cases.
(1) The reader may check that (1,0) and (0, ψ), x 12 x 11 + x 33 x 32 −x 13 x 23 are both elements in Ext 1 A−G (V 2 , V 2 ) considered in the Yoneda complex. (2) Writing up the syzygies we find that for i > j, See [6] for a detailed computation of all cases. Notice, however, that there does not yet exist a computer program computing this dimension (or invariants in general) under the action of an infinite group.