# Group actions, orbit spaces, and noncommutative deformation theory/Ruhmatoimed, orbiitruumid ja mittekommutatiivne deformatsiooniteooria.

1. 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 [member of] X corresponds to an A [G] -module [M.sub.x] = A/[a.sub.x], so that the classification of closures of orbits can be studied locally by deformation theory of [M.sub.x] as an A-G-module. Finally, we work through an example of the noncommutative blowup of cyclic surface singularities.

2. 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.sub.x] be the ideal of the closure of the orbit of x and let G [right arrow] [Aut.sub.k](A) sending g to [[nabla].sub.g] be the induced action of G on A. Then, as the ideal ax is invariant under the action of G on A, we get an induced action on A/ax. The skew group algebra over A is denoted A[G]. It consists of all formal sums [summation over (g[member of]G) [a.sub.g]g with product defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

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.sub.x] given by (ag)m = a[[nabla].sub.g](m) defines [M.sub.x] as an A[G]-module because

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus the classification of orbits is the classification of the corresponding A[G]-modules [M.sub.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

[[nabla].sub.g](am) = [[nabla].sub.g](a)[[nabla].sub.g](m).

Lemma 2.1. The category of A-G-modules and the category of A[G]-modules are equivalent.

3. DEFORMATION THEORY

For A a not necessarily commutative k-algebra, V = {[V.sub.i]}.sup.r.sub.r=1] a swarm of right A-modules (which means that [dim.sub.k][Ext.sup.1.sub.A]([V.sub.i]; [V.sub.j]) < [infinity] for 1 [less than ore equal to] i, j [less than or equal to] r), there exists a well-known deformation theory, see [3]. Let [a.ksub.r] be the category of r-pointed artinian k-algebras. It consists of the commutative diagrams

[ILLUSTRATION OMITTED]

such that rad(R) = ker(p) fulfills rad[(R).sup.n] = 0 for some n. Generalizing the commutative case, we set [[??].sub.r] equal to the category of complete r-pointed k-algebras [??] such that [??]/rad[([??]).sup.n] is in [a.sub.r] for all n. Letting [R.sub.ij] = [e.sub.i][Re.sub.j], it is easy to see that R is isomorphic to the matrix algebra ([R.sub.ij]). The noncommutative deformation functor [Def.sub.V] : [a.sub.r] [right arrow] Sets is given by

[Def.sub.v] (R) = {R [[cross product].sub.k] [A.sup.op]-modules [V.sub.r]|[V.sub.r] [[congruent to].sub.R] ([R.sub.ij] [[cross product].sub.k] [V.sub.j]), [k.sub.i] [[cross product].sub.R] [V.sub.R] [[congruent to].sub.R] [V.sub.i}/ [[congruent to].sub.R].

Let [V.sub.R] [member of] [Def.sub.V] (R). The left R-module structure is the trivial one, and the right A-module structure is given by the morphisms [[sigma].sup.R.sub.a] : [V.sub.i] [right arrow] [R.sub.ij] [[cross product].sub.k] [V.sub.j]. As in the commutative case, an (r-pointed) morphism [phi] : S [??] R is small if ker [phi] x rad(S) = rad(S) x ker [phi] = 0, and for such morphisms, lifting the [[sigma].sup.R] directly to S, the associativity condition gives the obstruction class o([phi], [V.sub.R]) = ([[sigma].sup.S.sub.ab] - [[sigma].sup.S.sub.b]) [member of] I [[cross product].sub.k] [HH.sup.2](A, [Hom.sub.k]([V.sub.i], [V.sub.j])) where I = ([I.sub.ij]) = ker [phi], such that [V.sub.R] can be lifted to [V.sub.S] if and only if o([V.sub.R], [phi]) = 0, see [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 V = [{[V.sub.i]}.sup.r.sub.i=1] of A-G-modules, we put

[Def.sup.G.sub.V](R) = {[V.sub.R] [member of] [Def.sub.V](R)|[there exists]A - G-structure [nabla] : G [right arrow] End ([V.sub.R])} [[subset].bar] [Def.sub.V](R).

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[[epsilon]] = k[[epsilon]]/[epsilon].sup.2] and use the tangent space

[Def.sub.V] (k[[epsilon]]) [congruent to] ([HH.sup.1] (A, [Hom.sub.k] ([V.sub.i], [V.sub.j]))) [congruent to] [Ext.sup.2.sub.A]([V.sub.i], [V.sub.j])

as dual basis for the local formal moduli [??]. The relations among the base elements are given by the obstruction space

[HH.sup.2] (A, [Hom.sub.k] ([V.sub.i], [V.sub.j])) = ([Ext.sup.2.sub.A] ([V.sub.i], [V.sub.j])).

4. GENERALIZED MATRIX MASSEY PRODUCTS (GMMP)

Let [{[V.sub.i]}.sup.r.sub.i=1] be a given swarm of A-modules. For each i, choose free resolutions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and we can prove Lemma 4.1 following the proof in [6] step by step:

Lemma 4.1. Let [V.sub.S] G [Def.sub.V](S) and let [phi] : R [right arrow] S be a small surjection. Then there exists a resolution [L.sup.S.sub..] = (S [[cross product].sub.k] L., [L.sup.S]) lifting the complex L., and to give a lifting [V.sub.R] of [V.sub.S] is equivalent to lift the complex [L.sup.S.sub..] to [L.sup.R.sub..]

Proof. Generalized from the commutative case, [M.sub.R] [[congruent to].sub.R] ([R.sub.ij] [[cross product].sub.k] [M.sub.j]) is equivalent with [M.sub.R] R-flat. Using this, and tensoring the sequence 0 [right arrow] I [right arrow] R [right arrow] S [right arrow] 0 with [M.sub.R] over R, gives the sequence 0 [right arrow] I [[cross product].sub.k] M [right arrow] [M.sub.R] [right arrow] [M.sub.S] [right arrow] 0. Ordinary diagram chasing then proves that the resolution of [M.sub.S] can be lifted to an R- complex [L..sup.R] given the resolution [L..sup.R] of [M.sub.S]. Conversely, given a lifting [L..sup.R] of the complex [L..sup.S] of [M.sub.S], the long exact sequence proves that this complex is a resolution, and that [M.sub.R] = [H.sup.0]([L..sup.R]) is a lifting of MS. []

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

[ILLUSTRATION OMITTED]

This proves that Lemma 4.1 is a particular case of the same lemma with [Def.sub.V](S) replaced by [Def.sup.G.sub.V](S). In [7] we give the definition of GMMP. The tangent space of the deformation functor is [Def.sup.G.sub.V](E) [congruent to] ([Ext.sup.1.sub.A-G]([V.sub.i], [V.sub.j])), where E is the noncommutative ring of dual numbers, i.e. E = k < [t.sub.ij] > /[([t.sub.ij]).sup.2]. For computations we note that when G is reductive and finite dimensional, [Hom.sub.A-G]([V.sub.i], [V.sub.j]) [congruent to] [Hom.sub.A][([V.sub.i], [V.sub.j]).sup.G] and [Ext.sup.1.sub.A-G]([V.sub.i], [V.sub.j]) = [Ext.sub.A][([V.sub.i], [V.sub.j]).sup.G], G acting by conjugation. Given a small surjection [phi] : R [right arrow] S, with kernel I = ([I.sub.ij]), lift [d..sup.S] on S [[cross product].sub.k] L. to [d..sup.R] on R [[cross product].sub.k] L. in the obvious way. Then o([phi], [V.sub.S]) = [{[d.sup.R.sub.i] [d.sup.R.sub.i-1]}.sub.i[greater than or equal to]1] [member of] ([I.sub.ij] [[cross product].sub.k] [Ext.sub.A-G]([V.sub.i], [V.sub.j])). By the definition of GMMP in [7], these can be read out of the coefficients of a basis in the obstruction space above.

5. THE MCKAY CORRESPONDENCE

Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

act on [A.sup.2.sub.C] by [tau](a, b) = (-a, -b). Our goal is to classify the G-orbits, and to find a compactification [[??].sub.G] [??][P.sup.2.sub.C] of the orbit space [M.sub.G]. The existing partial solution is

[M.sub.G] = Spec(k[[x.sup.2], xy, [y.sup.2]]) = Spec([A.sup.G]), A = k[x, y].

This is an orbit space, but not moduli. Consider the point P = (a, b) = ([square root of w, t], [square root of w]), w [not equal to] 0. Then

o(P) = {([square root of w, t] [square root of w]), ( -[square root of w], -t [square root of w])} = Z([I.sub.t]),

where [I.sub.t] = ([x.sup.2] - w, y - tx). We compute the local formal moduli of the A-G-module [M.sup.t] = A/[I.sub.t] from the diagram

[ILLUSTRATION OMITTED]

where the upper row is a resolution, we see that in general, [Ext.sup.A.sub.1] ([M.sub.t], [M.sub.t]) [congruent to] [Hom.sub.A] ([I.sub.t] / [I.sup.2.sub.t], A/[I.sub.t]) with the action of G given by conjugation, that is the composition given in the sequence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that [phi] = ([alpha], [beta]x) = [alpha](1, 0) + [beta](0, x) is invariant under the action of G. Writing this up in complex form, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We find [[xi].sub.1.sup.1] [[xi].sub.2.sup.1] = [[xi].sub.1.sup.2] [[xi].sub.2.sup.2] = [[xi].sub.1.sup.2] [[xi].sub.2.sup.2] = [[xi].sub.1.sup.1] [[xi].sub.2.sup.2] + [[xi].sub.1.sup.2] [[xi].sub.2.sup.1] = 0, which means that all cup-products are identically zero. Thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with algebraization [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Because the particular point [0.bar] = (0,0) corresponds to [M.sub.[0.bar]] = k[x, y]/(x, y) with [Ext.sup.1.sub.A-G]([M.sub.[0.bar]], [M.sub.[0.bar]]) = 0, we understand that [M.sub.[0.bar]] is a singular point, so that the modulus is [M.sub.G] = ([A.sup.2] - {[0.bar]}) [union] {pt}. At least in this case, resolving the singularity is a process of compactifying. Given a family V = [{[V.sub.i]}.sup.r.sub.i=1] of simple A-modules, an A-module E with composition series E = [E.sub.0] [contains] [E.sub.1] [contains] ... [contains] [E.sub.i] [contains] [E.sub.i-1] [contains] ... [contains] [E.sub.r] [contains] 0, where [E.sub.k] /[E.sub.k-1] = [V.sub.ik], is called an iterated extension of the family V, and the graph [GAMMA](E) of E (the representation type) is the graph with nodes in correspondence with V and arrows [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] connecting the nodes [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], identifying arrows if the corresponding extensions are equivalent. In [3] Laudal solves the problem of classifying all indecomposable modules E with fixed extension graph [GAMMA]. He proves that for every E there exists a morphism [PHI] : H(V) [right arrow] k[[GAMMA]] such that E [congruent to] [??] [[cross product].sub.[phi]] k[[GAMMA]], where [??] is the versal family, resulting in a noncommutative scheme Ind([GAMMA]). In [4], he then proves that the set [Simp.sub.n](A) of n-dimensional simple representations of A with the Jacobson topology has a natural scheme structure. He also proves that when [GAMMA] is a representation graph of dimension n = [[summation].sub.V[member of][GAMMA]] [dim.sub.k]V, then the set Simp([GAMMA]) = [Simp.sub.n](A) U Ind(r) has a natural scheme structure with the Jacobson topology, which is a compactification of [Simp.sub.n](A). In our present example, we let r be the representation type of the regular representation k[G]. We construct the composition series k[G] [congruent to] k[[tau]]/([[tau].sup.2] - 1) [contains] ([tau] - 1)/([[tau].sup.2] - 1) [contains] 0. Thus we get [V.sub.0] = k[[tau]]/([tau] - 1) [congruent to] k, [V.sub.1] = ([tau] - 1)/([[tau].sup.2] - 1) [congruent to] k and the action [[nabla].sup.i.sub.[tau]] of [tau] on [V.sub.i] is given by [[nabla].sup.i.sub.[tau]] = [(-1).sup.i] From the sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we immediately see that [Ext.sup.1.sub.A-G]([V.sub.i], [V.sub.j]) = [alpha](1, 0) + [beta](0, 1) when i [not equal to] j, 0 if i = j. Writing up the corresponding diagram and multiplying as in the previous example, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The versal family is given as the cokernel of the morphism

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now, as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] sends both [t.sub.21](1) and [t.sub.21](2) to 0. The isomorphism classes of indecomposable A [G]-modules with representation type r are thus given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The inherited group action is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on [k.sup.2]. To find Simp([GAMMA]), we start by computing the local formal moduli of the (worst) module [V.sub.t], following the algorithm in [2]. We find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by using (in particular) the fact that xy = yx in A. Then H[([V.sub.t]).sup.com] = k[v, w] with versal family [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], computed by again using the fact that xy = yx in A. While w = 0 gives the indecomposable module [V.sub.v+t], w [not equal to] 0 gives a simple two-dimensional A-G-module given by [x.sup.2] = w, xy = (t + v)w, [y.sup.2] = [(t + v).sup.2]w. This gives an embedding [A.sup.G] = k[[s.sub.0], [s.sub.1], [s.sub.2]]/([s.sub.0][s.sub.1] - [s.sup.2.sub.2]) = k[[x.sub.2], xy, [y.sub.2]] [??] k[v, w] inducing the morphism [Simp.sub.[GAMMA]] [right arrow] Spec([A.sub.G]) which is the ordinary blowup of the singular point. The exceptional fibre is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

doi: 10.3176/proc.2010.4.16

REFERENCES

[1.] Eriksen, E. An introduction to noncommutative deformations of modules. Lect. Notes Pure Appl. Math., 2005, 243(2), 90-126.

[2.] Laudal, O. A. Matric Massey products and formal moduli I. In Algebra, Algebraic Topology and Their Interactions (Roos, J.-E., ed.). Lecture Notes in Math., 1183, 218-240. Springer Verlag, 1986.

[3.] Laudal, O. A. Noncommutative deformations of modules. Homology Homotopy Appl., 2002, 4(2), 357-396.

[4.] Laudal, O. A. Noncommutative algebraic geometry. Rev. Mat. Iberoamericana, 2003, 19(2), 509-580.

[5.] Siqveland, A. The method of computing formal moduli. J. Algebra, 2001, 241 292-327.

[6.] Siqveland, A. Global Matric Massey products and the compactified Jacobian of the E6-singularity. J. Algebra, 2001, 241, 259-291.

[7.] Siqveland, A. A standard example in noncommutative deformation theory. J. Gen. Lie Theory Appl., 2008, 2(3), 251- 255.

Arvid Siqveland

Buskerud University College, PoBox 235, 3603 Kongsberg, Norway; arvid.siqveland@hibu.no

Received 14 April 2009, accepted 21 October 2009