Group actions, orbit spaces, and noncommutative deformation theory

Consider the action of a group G on an ordinary commutative k-variety X = Spec(A). In this note we define the category of A–G-modules and their deformation theory. We then prove that this deformation theory is equivalent to the deformation theory of modules over the noncommutative k-algebra A[G] = A]G. The classification of orbits can then be studied over a commutative ring, and we give an example of this on surface cyclic singularities.


INTRODUCTION
Consider the action of a group G on an ordinary commutative k-variety X = Spec(A).We define the category of A-G-modules, Definition 2.1, and their deformation theory.We then prove that this deformation theory is equivalent to the deformation theory of modules over the noncommutative k-algebra A[G] = A G. Thus the noncommutative moduli of the one-sided A[G]-modules can be computed as the noncommutative moduli of A-modules with A commutative, invariant under the (dual) action of the group G, which simplify the computations significantly.The orbit closure of x ∈ X corresponds to an A[G]-module M x = A/a x , so that the classification of closures of orbits can be studied locally by deformation theory of M x as an A-G-module.Finally, we work through an example of the noncommutative blowup of cyclic surface singularities.

MODULES WITH GROUP ACTIONS
Let k be an algebraically closed field of characteristic 0. Let G be a finite dimensional reductive algebraic group acting on an affine scheme X = Spec A, A a finitely generated (commutative) k-algebra.Let a x be the ideal of the closure of the orbit of x and let G → Aut k (A) sending g to ∇ g be the induced action of G on A. Then, as the ideal a x is invariant under the action of G on A, we get an induced action on A/a x .The skew group algebra over A is denoted A [G].It consists of all formal sums ∑ g∈G a g g with product defined by For later use notice that this definition extends the definition of the group algebra over k, k [G].Now, the action of A[G] on M x given by (ag)m = a∇ g (m) defines M x as an A[G]-module because Thus the classification of orbits is the classification of the corresponding A[G]-modules M x .The main issue of this section is the following definition and the lemma proved by the argument above: Definition 2.1.An A-G-module is an A module with a G-action such that the two actions commute, that is

The category of A-G-modules and the category of A[G]
-modules are equivalent.

DEFORMATION THEORY
, there exists a well-known deformation theory, see [3].Let a r be the category of r-pointed artinian k-algebras.It consists of the commutative diagrams such that rad(R) = ker(ρ) fulfills rad(R) n = 0 for some n.Generalizing the commutative case, we set âr equal to the category of complete r-pointed k-algebras R such that R/ rad( R) n is in a r for all n.Letting R i j = e i Re j , it is easy to see that R is isomorphic to the matrix algebra (R i j ).The noncommutative deformation functor Def V : a r → Sets is given by The left R-module structure is the trivial one, and the right A-module structure is given by the morphisms σ R a : As in the commutative case, an (r-pointed) morphism φ : S R is small if ker φ • rad(S) = rad(S) • ker φ = 0, and for such morphisms, lifting the σ R directly to S, the associativity condition gives the obstruction class o(φ [3] or [1] for details and complete proofs.Obviously, computations are much easier if A is a commutative k-algebra.This is possible to achieve when working with G-actions and orbit spaces.For a family In [2,3] Laudal constructs the local formal moduli of A-modules.In [5,6] applications in the commutative case are given, and in [7] an easy noncommutative example is worked through.In these cites we start with the k-algebra k as dual basis for the local formal moduli Ĥ.The relations among the base elements are given by the obstruction space HH 2 (A,

GENERALIZED MATRIX MASSEY PRODUCTS (GMMP)
Let {V i } r i=1 be a given swarm of A-modules.For each i, choose free resolutions 0 and we can prove Lemma 4.1 following the proof in [6] step by step: and let φ : R → S be a small surjection.Then there exists a resolution L S .= (S ⊗ k L., d S .) lifting the complex L., and to give a lifting V R of V S is equivalent to lift the complex L S .
Using this, and tensoring the sequence 0 Ordinary diagram chasing then proves that the resolution of M S can be lifted to an R-complex L. R given the resolution L. S of M S .Conversely, given a lifting L. R of the complex L. S of M S , the long exact sequence proves that this complex is a resolution, and that If M is an A-G-module where G acts rationally on A and M is a rational G-module, finitely generated as an A-module, then an A-free (projective) resolution of M can be lifted to an A-G-free resolution, that is a commutative diagram This proves that Lemma 4.1 is a particular case of the same lemma with Def V (S) replaced by Def G V (S).In [7] we give the definition of GMMP.The tangent space of the deformation functor is Def G , where E is the noncommutative ring of dual numbers, i.e.E = k < t i j > /(t i j ) 2 .For computations we note that when G is reductive and finite dimensional, , G acting by conjugation.Given a small surjection φ : R → S, with kernel . By the definition of GMMP in [7], these can be read out of the coefficients of a basis in the obstruction space above.

Let
Our goal is to classify the G-orbits, and to find a compactification MG → P 2 C of the orbit space M G .The existing partial solution is This is an orbit space, but not moduli.Consider the point P = (a, b) = ( √ w,t √ w), w = 0. Then where I t = (x 2 − w, y − tx).We compute the local formal moduli of the A-G-module M t = A/I t from the diagram 0 where the upper row is a resolution, we see that in general, Ext 1 with the action of G given by conjugation, that is the composition given in the sequence We get which means that all cup-products are identically zero.
A−G (M 0 , M 0 ) = 0, we understand that M 0 is a singular point, so that the modulus is M G = (A 2 − {0}) ∪ {pt}.At least in this case, resolving the singularity is a process of compactifying.Given a family V = {V i } r i=1 of simple A-modules, an A-module E with composition series called an iterated extension of the family V , and the graph Γ(E) of E (the representation type) is the graph with nodes in correspondence with V and arrows ρ i p ,i p+1 connecting the nodes V i p and V i p+1 , identifying arrows if the corresponding extensions are equivalent.In [3] Laudal solves the problem of classifying all indecomposable modules E with fixed extension graph Γ.He proves that for every E there exists a morphism φ : where M is the versal family, resulting in a noncommutative scheme Ind(Γ).In [4], he then proves that the set Simp n (A) of n-dimensional simple representations of A with the Jacobson topology has a natural scheme structure.He also proves that when Γ is a representation graph of dimension n = ∑ V ∈Γ dim k V , then the set Simp(Γ) = Simp n (A) ∪ Ind(Γ) has a natural scheme structure with the Jacobson topology, which is a compactification of Simp n (A).In our present example, we let Γ be the representation type of the regular representation k [G].We construct the composition series and the action ∇ i τ of τ on V i is given by ∇ i τ = (−1) i .From the sequence (x, y) Writing up the corresponding diagram and multiplying as in the previous example, we get .
The versal family is given as the cokernel of the morphism The inherited group action is ∇ τ = 1 0 0 −1 on k 2 .To find Simp(Γ), we start by computing the local formal moduli of the (worst) module V t , following the algorithm in [2].We find
y) = 0 w(t + v) v 0 by using (in particular) the fact thatxy = yx in A. Then H(V t ) com = k[v, w] with versal family x y −w −w(t + v) 1 −(t + v) x y, computed by again using the fact that xy = yx in A. While w = 0 gives the indecomposable module V v+t , w = 0 gives a simple two-dimensional A-G-module given by x 2 = w, xy = (t + v)w, y 2 = (t + v) 2 w.This gives an embeddingA G = k[s 0 , s 1 , s 2 ]/(s 0 s 1 − s 2 2 ) = k[x 2 , xy, y 2 ] → k[v, w] inducing the morphism Simp Γ → Spec(A G )which is the ordinary blowup of the singular point.The exceptional fibre is x y 0 0 −1 −t x y ∪V ∞ ∼ = P 1 .