Hilbert series of graded polynomial algebras

Let A = ⨁ₙ 𝒜 n be an ℕ-graded algebra over a commutative ring R, and let f i ∈ 𝒜 (d i) be homogeneous elements such that R[Xᵢ] → A, Xᵢ ↦ f i, is bijective: A is a polynomial algebra on the homogeneous generators f i. Giving the variable Xᵢ the weight d i, this evaluation is an isomorphism of graded algebras, so the degree-n piece 𝒜 n is the image of the weighted homogeneous polynomials of weight n, and it is free on the monomials ∏ᵢ Xᵢ ^ e i with ∑ᵢ e i * d i = n. When the generators are finitely many, of positive degrees, the Hilbert series of A is therefore

∑ₙ rank (𝒜 n) tⁿ = ∏ᵢ (1 - t ^ d i)⁻¹

(PowerSeries.mk_natCard_weight_mul_prod_one_sub_X_pow). Since this product determines the multiset of the positive integers d i (Multiset.eq_of_prod_map_one_sub_X_pow_eq), the multiset of degrees of a homogeneous polynomial generating system does not depend on the system: the generators of such a presentation are not canonical, but their degrees are. For a connected grading (𝒜 0 is the scalars) the degrees of such generators are automatically positive.

The main example is the algebra of invariant polynomials of a finite group generated by pseudoreflections, whose homogeneous polynomial generators are its basic invariants; their degrees are the degrees of the group.

Main results

• TauCeti.GradedAlgebra.aeval_weightedHomogeneousComponent: evaluation at homogeneous elements carries weighted homogeneous components to homogeneous components.
• TauCeti.GradedAlgebra.map_aeval_weightedHomogeneousSubmodule: the weighted homogeneous polynomials of weight n map onto 𝒜 n under a surjective evaluation at homogeneous elements.
• TauCeti.GradedAlgebra.finrank_eq_natCard_weight: for a polynomial presentation, Module.finrank (𝒜 n) equals Nat.card of the exponent vectors of weight n; when there are infinitely many such vectors both sides are 0.
• TauCeti.GradedAlgebra.ne_zero_of_injective_aeval: in a connected graded algebra, algebraically independent homogeneous elements have positive degrees.
• TauCeti.GradedAlgebra.mk_finrank_mul_prod_one_sub_X_pow: the Hilbert series of a graded polynomial algebra on generators of positive degrees d i is ∏ᵢ (1 - t ^ d i)⁻¹.
• TauCeti.GradedAlgebra.map_univ_eq_of_bijective_aeval: any two homogeneous polynomial generating systems of positive degrees have the same multiset of degrees.

References

• R. P. Stanley, Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. (N.S.) 1 (1979), 475–511, Section 3.
• J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press (1990), Section 3.7.

theorem TauCeti.GradedAlgebra.aeval_weightedHomogeneousComponent {R : Type u_1} {A : Type u_2} {ι : Type u_3} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {f : ι → A} {d : ι → ℕ} (hf : ∀ (i : ι), f i ∈ 𝒜 (d i)) (φ : MvPolynomial ι R) (n : ℕ) : (MvPolynomial.aeval f) ((MvPolynomial.weightedHomogeneousComponent d n) φ) = ↑(((DirectSum.decompose 𝒜) ((MvPolynomial.aeval f) φ)) n)

Evaluation at homogeneous elements is graded. If f i ∈ 𝒜 (d i), then evaluating at f the weighted homogeneous component of weight n of a polynomial φ, for the weights d, gives the degree-n component of the evaluation of φ.

theorem TauCeti.GradedAlgebra.map_aeval_weightedHomogeneousSubmodule {R : Type u_1} {A : Type u_2} {ι : Type u_3} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {f : ι → A} {d : ι → ℕ} (hf : ∀ (i : ι), f i ∈ 𝒜 (d i)) (hsurj : Function.Surjective ⇑(MvPolynomial.aeval f)) (n : ℕ) : Submodule.map (MvPolynomial.aeval f).toLinearMap (MvPolynomial.weightedHomogeneousSubmodule R d n) = 𝒜 n

If evaluation at homogeneous elements f i ∈ 𝒜 (d i) is surjective, it maps the polynomials that are weighted homogeneous of weight n for the weights d onto 𝒜 n.

theorem TauCeti.GradedAlgebra.finrank_eq_natCard_weight {R : Type u_1} {A : Type u_2} {ι : Type u_3} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {f : ι → A} {d : ι → ℕ} [Nontrivial R] (hf : ∀ (i : ι), f i ∈ 𝒜 (d i)) (hbij : Function.Bijective ⇑(MvPolynomial.aeval f)) (n : ℕ) : Module.finrank R ↥(𝒜 n) = Nat.card { e : ι →₀ ℕ // (Finsupp.weight d) e = n }

Dimensions of a graded polynomial algebra. If evaluation at homogeneous elements f i ∈ 𝒜 (d i) is bijective, then Module.finrank (𝒜 n) equals Nat.card of the exponent vectors e of weight ∑ᵢ e i * d i = n. When there are infinitely many such vectors, 𝒜 n has an infinite basis and both sides are 0.

theorem TauCeti.GradedAlgebra.ne_zero_of_injective_aeval {R : Type u_1} {A : Type u_2} {ι : Type u_3} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) {f : ι → A} {d : ι → ℕ} [Nontrivial R] (h𝒜 : ∀ x ∈ 𝒜 0, x ∈ ⊥) (hf : ∀ (i : ι), f i ∈ 𝒜 (d i)) (hinj : Function.Injective ⇑(MvPolynomial.aeval f)) (i : ι) : d i ≠ 0

Generators of a connected graded algebra have positive degrees. If the degree-zero part of A consists of scalars, then a homogeneous element of degree zero is not algebraically independent of the others: if evaluation at f i ∈ 𝒜 (d i) is injective, every d i is nonzero.

theorem TauCeti.GradedAlgebra.mk_finrank_mul_prod_one_sub_X_pow {R : Type u_1} {A : Type u_2} {ι : Type u_3} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {f : ι → A} {d : ι → ℕ} [Fintype ι] [Nontrivial R] (S : Type u_4) [CommRing S] (hf : ∀ (i : ι), f i ∈ 𝒜 (d i)) (hbij : Function.Bijective ⇑(MvPolynomial.aeval f)) (hd : ∀ (i : ι), d i ≠ 0) : (PowerSeries.mk fun (n : ℕ) => ↑(Module.finrank R ↥(𝒜 n))) * ∏ i : ι, (1 - PowerSeries.X ^ d i) = 1

The Hilbert series of a graded polynomial algebra. If evaluation at homogeneous elements f i ∈ 𝒜 (d i) of positive degrees is bijective, the Hilbert series ∑ₙ rank (𝒜 n) tⁿ of A (with coefficients cast into any commutative ring S) is the inverse of ∏ᵢ (1 - t ^ d i).

theorem TauCeti.GradedAlgebra.map_univ_eq_of_bijective_aeval {R : Type u_1} {A : Type u_2} {ι : Type u_3} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {f : ι → A} {d : ι → ℕ} [Fintype ι] [Nontrivial R] {κ : Type u_4} [Fintype κ] {f' : κ → A} {d' : κ → ℕ} (hf : ∀ (i : ι), f i ∈ 𝒜 (d i)) (hbij : Function.Bijective ⇑(MvPolynomial.aeval f)) (hd : ∀ (i : ι), d i ≠ 0) (hf' : ∀ (j : κ), f' j ∈ 𝒜 (d' j)) (hbij' : Function.Bijective ⇑(MvPolynomial.aeval f')) (hd' : ∀ (j : κ), d' j ≠ 0) : Multiset.map d Finset.univ.val = Multiset.map d' Finset.univ.val

The degrees of a graded polynomial algebra are intrinsic. If A is a polynomial algebra on homogeneous generators f i ∈ 𝒜 (d i) and also on homogeneous generators f' j ∈ 𝒜 (d' j), all of positive degree, then the two families have the same multiset of degrees.