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Geometry of Noncommutative k-Algebras | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Geometry of Noncommutative k-Algebras

Arvid Siqveland*

Buskerud University College, Department of Technology, P.O. Box 251, N-3601 Kongsberg, Norway

*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



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Abstract

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.

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 DefV has a (unique up to nonunique isomorphism) prorepresenting hull (local formal moduli) equation isomorphic to the completed local ring, that is equation 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 : kr → R and g : R → kr 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 = {V1, . . . , Vr} of right A-modules, there is a noncommutative deformation functor DefV : ar → Sets. If Ext1A(Vi, Vj ) has a finite k-dimension for 1 ≤ i, j ≤ r, Laudal (or equally Eriksen) proves that DefV has a formal moduli (equation), 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 = {V1, . . . , Vr} be the family of simple right A-modules. Then, the (equation-flat) proversal family equationis an isomorphism equation

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 m1, m2 two different maximal ideals with corresponding simple modules Vi = A/mi, i = 1, 2. Then, Ext1A(V1, V2) = 0.

Proof. It is enough to consider

equation

with α1 ≠0. First of all, it is well known that

equation

The inner derivations are given by equationin this case, and this determines the (inner) derivations completely. Now, let δ : A → Hom(V1, V2) be a derivation. Then, since A is commutative,

equation

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

equation

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

equation

let

equation

We consider the two-sided ideal in F generated by f11, that is a = ⟨f11⟩, and for the quotient algebra we use the notation

equation

In this case Q = (Qij ), and k⟨t11(1), t11(2)⟩ maps injective into Q, but Q11 = k⟨t11(1), t11(2)⟩ as for example t12t21 ∈ Q11. However, letting ⟨Q − Qii⟩ be the ideal generated by the matrices in Q with 0 (i, i)-entry, we will write ⟩Q11 = k⟨t11(1), t11(2)⟩ = Q/⟨Q − Q11⟩ when necessary.

Let kr → R = (Rij ) be a matrix algebra. We let ⟨R − Rii⟩ denote the ideal generated by the matrices in R with 0 (i, i)-entry, and we let equation denote the quotient R/⟨R − Rii⟩. We call the algebras equation the diagonal algebras of the matrix algebra R = (Rij ). We let equation be the canonical morphism, and we let equation be the natural inclusion. Then, τii obeys the rules for an algebra morphism except for the fact that τii(1)&ne 1. Thus, τii−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, equation because 1 ∈ m otherwise. We see that for m ∈ m, τiiii(m)) ∈ m implying that ιii(m) ∈ τii−1 (m) so that m ⊆ ιii−1ii−1 (m)). Because mis maximal, m = ιii−1ii−1 (m)) and τii−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

equation

such that ρL(a) is a unit in L whenever κA(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 → Endk(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 = ρa 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 = {V1, . . . , Vn} be simple right A-modules. Then, a k-algebra L is called a localization of A in V if there exists a diagram

equation

such that ρL(a) is a unit in L whenever κAi(a) is a unit in Homk(Vi, Vi) 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 m1, . . . ,mn be maximal ideals. Pu t Vi = A/mi, 1 ≤ i ≤ n. Then, equation fulfils the condition of being a localization of A in V = {Vi, . . . , Vn}. Notice that the set of simple modules of L are the modules V .

Example 9. Let A be any k-algebra and V1, . . . , Vn simple right A-modules. Assume that there exists a k-algebra equation such that each Li is finitely generated with Vi as the only simple Limodule. Also assume that equation and that Li is miniversal (in the meaning that Li is an algebraizationof equation. Then L≅AV , 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 = {V1, . . . , Vn.} a family of simple right A-modules. Then, L is called a prolocalization of A in V if there exist diagrams

equation

for each i, 1 ≤ i ≤ n, such that ρL(a) is a unit in L whenever κAi(a) is a unit for each i, and if for each i one hasequation. One writes equation and notices that prolocalizations are not unique.

Lemma 11.Prolocalizations exist.

Proof. Note that equation satisfies the properties of the definition. The homomorphism equation is surjective and equation is a unit whenever equation is a unit in equation implying that{V1, . . . , Vn} is exactly the set of simple L-modules. Now, let equation. Then, bythe generalized Burnsides theorem, Theorem 2, we have the matrix algebra equation,implying in particular that equation. Taking the projective limit, we then end atequation for each i, proving the claim.

Lemma 12. Let V = {V1, . . . , Vn} be a set of simple right A-modules. Let equation be the prolocalization of A in V . Then, Simp(equation ) = V .

Proof. Note that equationmaps surjectively onto Endk(Vi), so by Lemma 6, Vi is a simpleequation-module, that is V ⊆ Simp( equation). It is also obvious that ifequation maps to a unit inequation it is itself a unit. Thus equation 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 = {V1, . . . , Vn} a family of simple right A-modules. Then, L is called a geometric prolocalization of A in V if there exists diagrams

equation

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

equation

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

Lemma 14. The geometric prolocalization equation of A in V = {V1, . . . , Vn} exists, and equation

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

equation

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

equation

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

(2) if A is commutative, then equation

Proof. (1) We see that A≅A1 and so this follows by definition.

(2) This follows as

equation

Definition 16. We call (SimpA,equation) 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 equation of A-modules, not necessarily finite. On the set |c|, we define the Jacobson topology generated by the open subsets equation for f ∈ A given by equation where equation is the structure morphism and where equation denotes the units in this kalgebra. We let equation when V = {V1, . . . , Vn}.

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

equation

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

equation

Then, equation 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 equation

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

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

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

equation

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 GL3(k)-orbits of M3(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 M3(k) = Spec(A), A = k[xij ]1 ≤ i ,j ≤ 3, then G := GL3(k) acts on A by conjugacy, that is, g = (αij ) ∈ G acts linearly on A by g(xij) = (αij )(xij )(αij)−1. Denote this action by ∇ : G → Aut(A). Let M be an A-module, and let ∇ : G → Aut(M) be an action such that

equation

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

equation

(2) Two different eigenvalues

The following Jordan forms are possible, letting equation:

equation

The case

equation

will be correspondingly.

(3) Three different eigenvalues

The one and only orbit is the orbit of

equation

Now we set Mλ = M − λI,

equation

For simplicity, we will also use the notation

equation

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

Lemma 20. equation

Proof. equation Thus equation

In the case with exactly two different eigenvalues λ1≠ λ2 and λ = (λ1, λ2), the orbit closures are given by the following.

Lemma 21. equation

Proof. From direct computation:

equation

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

Proposition 22. The k-dimension of equation is given as the (i, j) entry in the matrix

equation

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),

equation

are both elements in equation considered in the Yoneda complex.

(2) Writing up the syzygies we find that for i > j,

equation

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 equation denote the formal local noncommutative moduli of the modules equation 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 equation be the automorphism sending xij to equation . This automorphism sends si to equation to equation , 1 ≤ i, j ≤ 3. Because φλ obviously commutes with the group action, that is, because the diagram

equation

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

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

Proof. The automorphism φλ transforms every computation with tangent space bases, resolutions and Massey products for equation

And in the case with two coinciding eigenvalues as follows.

Lemma 24. For every equation one has that

equation

Proof. Use the automorphism equation described in the previous section. This automorphism sends s1 −(λ2 − λ1) to equation s2 to equation s3 to equation and sij to equation 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 V1, V2, V3 corresponding to one eigenvalue, V1, V2 corresponding to two different eigenvalues and V corresponding to three different eigenvalues.

Proposition 25. Let

equation

Then, the noncommutative local formal moduli of the modules corresponding to the closure of the orbits of the Jordan forms M1, M2, M3 is

equation

where b is the ideal generated by

equation

equation

Proposition 26. Let equation

Then, the noncommutative local formal moduli of the modules corresponding to the closure of the orbits of M1 and M2 is

equation

where

equation

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

equation

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

equation

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

equation

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 equation correspond to the points on the surface

equation

The forms equationwith coinciding eigenvalues give the curve

equation

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 equation to be the affine ring for M3(k)/GL3(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 equationwe can compute the tangent space dimensions equation by looking at k-derivations δ. The dimension drops if δ(f) = 0 for some relation f. Let equation and equation be three points on the diagonal of M. Then, the constant equation -locus is given as follows:

(1, 2)

equation

We put

equation

and we get the equations

equation

which is exactly the point

equation

on the surface. (1, 3)

equation

We put

equation

and we get the following equations:

equation

This gives the points on the curve

equation (2, 3)

equation

On the curve, the above chosen parameters correspond to

equation

that is

equation

This is true for both equations above:

equation

Thus, the constant ext1-locus is preserved on the curve.

The constant ext1-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)

equation

We let

equation

Then, we get the equations

equation

which are the surface

equation

This gives the picture of the moduli for GL3(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.

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