The invariant ring of a finite linear group at its vertex

Let a finite group G act linearly on a module M over a commutative ring R, and so on the symmetric algebra S = Sym M, with invariant subalgebra A = Sym(M)^G. Write S₊ for the irrelevant ideal of S and A₊ = S₊ ∩ A for the ideal of invariants with zero constant term, the vertex of the quotient Spec S → Spec A. This file compares A at A₊ with S at S₊.

Over a domain, S₊ is the only prime of S lying over A₊: the primes over A₊ form a single G-orbit, and S₊ is G-stable. Geometrically, the fibre of Spec S → Spec A over the vertex has a single point, the origin.

When M is finitely generated, S is a finite A-module, and A is of finite type over a Noetherian base (Noether's finiteness theorem, see TauCeti.RepresentationTheory.SymmetricAlgebra.FiniteType). Over a field k with M finite-dimensional, prime heights are then unchanged under contraction from S to A, so the vertex has height finrank k M and the local ring A_{A₊} has Krull dimension finrank k M, as does A itself. For M = V* this says that the quotient V/G has dimension dim V at its vertex, the dimension count needed to identify the invariant ring of a group generated by pseudoreflections with a polynomial ring.

Main results

• Representation.ringKrullDim_symmetricAlgebraInvariants: the invariant subalgebra has the Krull dimension of Sym M.
• Representation.primesOver_under_irrelevant_eq_singleton: over a domain, S₊ is the only prime of Sym M lying over A₊.
• Representation.height_under_symmetricAlgebraInvariants: over a field, contraction to the invariants preserves the heights of ideals.
• Representation.height_under_irrelevant and Representation.ringKrullDim_atPrime_under_irrelevant: over a field, A₊ has height finrank k M, so A_{A₊} has Krull dimension finrank k M.

References

• H. Derksen, G. Kemper, Computational Invariant Theory, Springer (2002), Chapter 2.
• L. Smith, Polynomial Invariants of Finite Groups, A K Peters (1995), Chapter 2.

theorem Representation.ringKrullDim_symmetricAlgebraInvariants {R : Type u_1} {G : Type u_3} {M : Type u_4} [CommRing R] [Group G] [Finite G] [AddCommMonoid M] [Module R M] (ρ : Representation R G M) : ringKrullDim ↥ρ.symmetricAlgebraInvariants = ringKrullDim (SymmetricAlgebra R M)

For a finite group, the invariant subalgebra has the Krull dimension of the symmetric algebra, which is integral over it.

theorem Representation.primesOver_under_irrelevant_eq_singleton {R : Type u_1} {G : Type u_3} {M : Type u_4} [CommRing R] [Group G] [Finite G] [AddCommMonoid M] [Module R M] (ρ : Representation R G M) [IsDomain R] : (Ideal.under (↥ρ.symmetricAlgebraInvariants) (HomogeneousIdeal.irrelevant (TauCeti.SymmetricAlgebra.homogeneousSubmodule R M)).toIdeal).primesOver (SymmetricAlgebra R M) = {(HomogeneousIdeal.irrelevant (TauCeti.SymmetricAlgebra.homogeneousSubmodule R M)).toIdeal}

The fibre over the vertex has a single point. Over a domain, the irrelevant ideal S₊ is the only prime of Sym M lying over the ideal A₊ of invariants with zero constant term.

theorem Representation.eq_irrelevant_of_liesOver {R : Type u_1} {G : Type u_3} {M : Type u_4} [CommRing R] [Group G] [Finite G] [AddCommMonoid M] [Module R M] (ρ : Representation R G M) [IsDomain R] (P : Ideal (SymmetricAlgebra R M)) [P.IsPrime] [P.LiesOver (Ideal.under (↥ρ.symmetricAlgebraInvariants) (HomogeneousIdeal.irrelevant (TauCeti.SymmetricAlgebra.homogeneousSubmodule R M)).toIdeal)] : P = (HomogeneousIdeal.irrelevant (TauCeti.SymmetricAlgebra.homogeneousSubmodule R M)).toIdeal

Over a domain, the only prime of Sym M lying over the ideal A₊ of invariants with zero constant term is the irrelevant ideal S₊.

theorem Representation.height_under_symmetricAlgebraInvariants {k : Type u_2} {G : Type u_3} {M : Type u_4} [Field k] [Group G] [Finite G] [AddCommGroup M] [Module k M] [FiniteDimensional k M] (ρ : Representation k G M) (P : Ideal (SymmetricAlgebra k M)) : (Ideal.under (↥ρ.symmetricAlgebraInvariants) P).height = P.height

Over a field, for a finite group acting on a finite-dimensional space M, contracting an ideal of Sym M to the invariant subalgebra preserves its height.

theorem Representation.height_under_irrelevant {k : Type u_2} {G : Type u_3} {M : Type u_4} [Field k] [Group G] [Finite G] [AddCommGroup M] [Module k M] [FiniteDimensional k M] (ρ : Representation k G M) : (Ideal.under (↥ρ.symmetricAlgebraInvariants) (HomogeneousIdeal.irrelevant (TauCeti.SymmetricAlgebra.homogeneousSubmodule k M)).toIdeal).height = ↑(Module.finrank k M)

The dimension of the invariant ring at its vertex. Over a field, for a finite group acting on a finite-dimensional space M, the ideal A₊ of invariants with zero constant term has height finrank k M.

theorem Representation.ringKrullDim_atPrime_under_irrelevant {k : Type u_2} {G : Type u_3} {M : Type u_4} [Field k] [Group G] [Finite G] [AddCommGroup M] [Module k M] [FiniteDimensional k M] (ρ : Representation k G M) : ringKrullDim (Localization.AtPrime (Ideal.under (↥ρ.symmetricAlgebraInvariants) (HomogeneousIdeal.irrelevant (TauCeti.SymmetricAlgebra.homogeneousSubmodule k M)).toIdeal)) = ↑(Module.finrank k M)

Over a field, for a finite group acting on a finite-dimensional space M, the localization of the invariant subalgebra at the ideal A₊ of invariants with zero constant term has Krull dimension finrank k M.