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- *Corresponding Author:
- Arvid Siqveland

Buskerud University College

Department of Technology

P.O. Box 251, N-3601 Kongsberg

Norway

**E-mail:**[email protected]

**Received date: ** 1 October 2009; **Accepted date: ** 26 January 2011

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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 OˆX,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 etal´e 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.

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) isomorphic to the completed local ring, that is 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.** a_{r} 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}, . . . , V_{r}} of right A-modules, there is a noncommutative deformation functor Def_{V} :
a_{r} → Sets. If Ext^{1}_{A}(V_{i}, V_{j} ) has a finite k-dimension for 1 ≤ i, j ≤ r, Laudal (or equally Eriksen) proves that Def_{V} has a formal moduli (), 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}, . . . , V_{r}}
be the family of simple right A-modules. Then, the (-flat) proversal family is an
isomorphism

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.

**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 modules V_{i} = A/m_{i}, i = 1, 2. Then, Ext^{1}_{A}(V_{1}, V_{2}) = 0.

Proof. It is enough to consider

with α_{1} ≠0. First of all, it is well known that

The inner derivations are given by 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.

**Example 4.** Consider the following matrix variables

The free 2 × 2 matrix k-algebra generated by these elements by ordinary matrix multiplication is then denoted

let

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 denote the quotient R/⟨R − Rii⟩. We call the algebras the diagonal algebras
of the matrix algebra R = (R_{ij} ). We let be the canonical morphism, and we let be the natural inclusion. Then, τ_{ii} obeys the rules for an algebra morphism except for the fact that
τ_{ii}(1)&ne 1. Thus, τ_{ii}^{−1} (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, because 1 ∈ m otherwise.
We see that for m ∈ m, τ_{ii}(ι_{ii}(m)) ∈ m implying that
ι_{ii}(m) ∈ τ_{ii}^{−1} (m) so that m ⊆ ι_{ii}^{−1} (τ_{ii}^{−1} (m)). Because mis maximal, m = ι_{ii}^{−1} (τ_{ii}^{−1} (m)) and τ_{ii}^{−1} (m) 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

such that _{ρL}(a) is a unit in L whenever κ_{A}(a) is a unit in A/m. For any other L^{'} with this property, there exists a unique morphism φ : L → L _{'} such that ρL^{'} = ρL ◦ φ.

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.

**Lemma 6.** V is a simple A module if the structure morphism ? : A → End_{k}(V ) is surjective. If k is algebraically
closed, the converse holds.

Proof. LetW be a submodule of V , let 0 ≠ ω ∈ W be an element, and let υ ∈ 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 = ρ_{a} for some a ∈ A
because of the surjectivity. Then, υ = φ(ω) = a · ω ∈ 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 V = {V_{1}, . . . , V_{n}} be simple right
A-modules. Then, a k-algebra L is called a localization of A in V if there exists a diagram

such that ρL(a) is a unit in L whenever κ^{A}_{i}(a) is a unit in Hom_{k}(V_{i}, V_{i}) for every i, 1 ≤ i ≤ n, and if for any other L' with this property, there exists a unique φ : L → L' such that ρL' = ρL ◦ φ.

**Example 8.** As an elementary example, let A be commutative and let m_{1}, . . . ,m_{n} be maximal ideals. Pu t V_{i} = A/m_{i}, 1 ≤ i ≤ n. Then, fulfils the condition of being a localization of A in V = {V_{i}, . . . , V_{n}}. Notice that the set of simple modules of L are the modules V .

**Example 9.** Let A be any k-algebra and V_{1}, . . . , V_{n} simple right A-modules. Assume that there exists a k-algebra such that each L_{i} is finitely generated with V_{i} as the only simple Limodule. Also assume that and that Li is miniversal (in the meaning that Li is an algebraizationof . Then L≅A_{V} , the localization of A in the family V .

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}, . . . , V_{n}.} a family of simple right A-modules. Then, L is called
a prolocalization of A in V if there exist diagrams

for each i, 1 ≤ i ≤ n, such that ρL(a) is a unit in L whenever κ^{A}_{i}(a) is a unit for each i, and if for each i one has. One writes and notices that prolocalizations are not unique.

**Lemma 11.***Prolocalizations exist.*

Proof. Note that satisfies the properties of the definition. The homomorphism is surjective and is a unit whenever is a unit in implying that{V_{1}, . . . , V_{n}} is exactly the set of simple L-modules. Now, let . Then, bythe generalized Burnsides theorem, Theorem 2, we have the matrix algebra ,implying in particular that . Taking the projective limit, we then end at for each i, proving the claim.

**Lemma 12.** Let V = {V_{1}, . . . , V_{n}} be a set of simple right A-modules. Let be the prolocalization of A in V .
Then, Simp( ) = V .

Proof. Note that maps surjectively onto End_{k}(V_{i}), so by Lemma 6, V_{i} is a simple-module, that is V ⊆ Simp( ). It is also obvious that if maps to a unit in it is itself a unit. Thus 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}, . . . , V_{n}} a family of simple right
A-modules. Then, L is called a geometric prolocalization of A in V if there exists diagrams

for each i, 1 ≤ i ≤ n, such that ρL(a) is a unit in L whenever κ^{A}_{i}(a) is a unit for each i, and if there exists an isomorphism of matrix k-algebras

We write , and notice that geometric prolocalizations are not unique.

**Lemma 14.** The geometric prolocalization of A in V = {V_{1}, . . . , V_{n}} exists, and

Proof. Put . Then exactly as above, fulfils the conditions. Notice that even for anoncommutative k-algebra, (u + f)(p − pfp + pfpfp − pfpfpfp + · · · ) = 1 when and p is a rightunit of υ (we recall that rad( ) = ker η, where 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:

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 define D(f) = {V ∈ SimpA | ρ(f) : V → V is invertible}, where ρ : A → Endk(V ) is the structure morphism. We have D(f)D(g) = D(fg), and so we can let the topology on SimpA be the topology with base of open subsets D(f), f ∈ A.

For f ∈ A, we define

We then define the sheaf of regular, not necessarily commutative, functions on X = SimpA by

Now if all the are algebraizable, that is, there exist algebraic localizations of A for every finite subset c with natural and coherent morphisms for each inclusion , 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:*

(1)

(2) if A is commutative, then

Proof. (1) We see that A≅A_{1} and so this follows by definition.

(2) This follows as

**Definition 16.** We call (SimpA,) an affine scheme, and we say that the set of simple A-modules | SimpA|
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 of A-modules, not necessarily finite. On the *set |c|*, we define the Jacobson topology
generated by the open subsets for f ∈ A given by where is the structure morphism and where denotes the units in this kalgebra.
We let when V = {V_{1}, . . . , V_{n}}.

Then, we define a sheaf of r-pointed k-algebras on the topological space |c| as follows. At first, let P(U) = Then, we define

Notice that if is finite, This follows directly from the definition. Given this, we now define

Then, 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 is finite, 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, is a (not necessarily commutative) scheme. Moreover, the natural morphism glues together to a global module on . By the geometric properties, it is reasonable to call a moduli for , the original set of A-modules.

**Definition 18**. c is called an affine scheme for the k-algebra A if

**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
M_{3}(k) = Spec(A), A = k[x_{ij} ]1 ≤ i ,j ≤ 3, then G := GL_{3}(k) acts on A by conjugacy, that is, g = (α_{ij} ) ∈ G
acts linearly on A by g(x_{ij}) = (α_{ij} )(x_{ij} )(α_{ij})−1. Denote this action by ∇ : G → Aut(A). Let M be an A-module, and let ∇ : G → Aut(M) be an action such that

Then, (M,∇) is called an A − G-module. The category of A − G-modules is equivalent to the category of A[G]- modules, where A[G] is the skew group ring.

The affine k-algebra of the closure of a G-orbit is, by definition, an A−G-module of the form A/a where a is a G-stable ideal of A, together with the natural G-action:

Spec(A/a) ⊂ Spec(A).

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 :

The case

will be correspondingly.

(3) *Three different eigenvalues*

The one and only orbit is the orbit of

Now we set M^{λ} = M − λI,

For simplicity, we will also use the notation

We then have the following representation of the ideals defining the closures.

**Lemma 20.**

Proof. Thus

In the case with exactly two different eigenvalues λ_{1}≠ λ_{2} and λ = (λ_{1}, λ_{2}), the orbit closures are given by the following.

**Lemma 21.**

**Proof.** From direct computation:

And of course, in the case with three different eigen values the orbit closure is given by

**Proposition 22.** The k-dimension of 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),

are both elements in 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.

**The local formal moduli**

Let denote the formal local noncommutative moduli of the modules corresponding to the closures of the orbits. We will compute this k-algebra in the worst case situation, which is seen to be the case where all eigenvalues are equal, and three closures are contained in each other: the generic case.

Let be the automorphism sending xij to . This automorphism sends si to to , 1 ≤ i, j ≤ 3. Because φ^{λ} obviously commutes with the group action, that is, because the diagram

obviously commutes, we get the following, first in the case with three coinciding eigenvalues as follows.

**Lemma 23.** *For every λ ∈ k, let* Then

Proof. The automorphism φ^{λ} transforms every computation with tangent space bases, resolutions and Massey
products for

And in the case with two coinciding eigenvalues as follows.

**Lemma 24.** *For every* *one has that*

Proof. Use the automorphism described in the previous section. This automorphism sends s1 −(λ_{2} − λ_{1}) to s_{2} to s_{3} to and s_{ij} to for 1 ≤ i, j ≤ 3. Thus, the tangent spaces, the resolutions and the computation of Massey Products are isomorphic.

The computations of the local formal moduli are based on resolutions of the A−G-modules and liftings of these. The representation of the Massey products given by obstructions are given previously in [7], the full details in [6].

Because of the lemmas above, we can write up the local formal moduli of every situation V_{1}, V_{2}, V_{3} corresponding
to one eigenvalue, V_{1}, V_{2} corresponding to two different eigenvalues and V corresponding to three different
eigenvalues.

**Proposition 25.** *Let*

*Then, the noncommutative local formal moduli of the modules corresponding to the closure of the orbits of the
Jordan forms M _{1}, M_{2}, M_{3} is*

where b is the ideal generated by

**Proposition 26.** Let

Then, the noncommutative local formal moduli of the modules corresponding to the closure of the orbits of M1
and M_{2} is

*where*

Now, it is also obvious that in the case with three different eigenvalues, the local formal moduli is

All the relations defining the local formal moduli are polynomials and the choice of defining systems in the computation of this polynomials, the proversal family, is algebraizable (see, e.g., [6]). Thus, we may replace the double brackets with simple brackets and let

Then, M = T/b together with the universal family

is a moduli for the orbit closures. This follows from Propositions 25 and 26 proving that the restriction to subdiagrams are correct, and from Lemmas 23 and 24 which prove that the family above is universal. Finally, we also need to prove that the points of this k-algebra corresponds to the orbit closures. This will follow from a study of the geometry.

**The geometry**

The endomorphisms with Jordan form correspond to the points on the *surface*

The forms with coinciding eigenvalues give the *curve*

The geometric picture should show three generic points. The case with all three eigenvalues different is well
known to be parameterized by the *points* in affine 3-space. A *point* in this affine 3-space, on the *surface*, represents
a new 3-dimensional affine space glued onto this point. A *point* on the *curve* on the *surface* represents a new 3-
dimensional affine space which is glued onto the point. Outside the curve and the surface, all points are identified.

Necessary conditions for the k-algebra to be the affine ring for M_{3}(k)/GL_{3}(k) are that the simple modules of this ring are in one-to-one correspondence with the orbits, and that it is closed under forming
local formal moduli for finite subsets of the simple modules. In particular, the Ext1-dimensions must coincide, and
the universal family must exist.

Recalling (again) that we can compute the tangent space dimensions by looking at k-derivations δ. The dimension drops if δ(f) = 0 for some relation f. Let and be three points on the diagonal of M. Then, the constant -locus is given as follows:

(1, 2)

We put

and we get the equations

which is exactly the point

on the surface. (1, 3)

We put

and we get the following equations:

This gives the points on the curve

(2, 3)

On the curve, the above chosen parameters correspond to

that is

This is true for both equations above:

Thus, the constant ext^{1}-locus is preserved on the curve.

The constant ext^{1}-locus for the local formal moduli for a point on the surface, that is the case with exactly two
different eigenvalues, is given by the equations (for simplicity we put λ = 1)

We let

Then, we get the equations

which are the surface

This gives the picture of the moduli for GL_{3}(k) as the affine 3-space, the affine 2-space and the curve and proves
the main theorem of the section. Notice that the affine 2-space in the middle is the blowup of the surface along the
curve.

- Eriksen E (2006) An introduction to noncommutative deformations of modules, Noncommutative Algebra and Geometry. Lect Notes Pure Appl Math, Chapman & Hall/CRC, Boca Raton, FL 243: 90–125.
- Lam TY (2001) A first course in noncommutative rings. Graduate Texts in Mathematics, Springer-Verlag, New York.
- Laudal OA (2002) Noncommutative deformations of modules. Homology Homotopy Appl 4:357–396.
- Laudal OA (2003) Noncommutative algebraic geometry. Rev Mat Iberoamericana 19: 509–580.
- Schlessinger M (1968) Functors of Artin rings. Trans Amer Math Soc 130: 208–222.
- Siqveland (2001) The noncommutative moduli of rk 3 endomorphisms. Report Series Buskerud University College 26 1–132.
- Siqveland (2008) A standard example in noncommutative deformation theory. J Gen Lie Theory Appl 2: 251–255.

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