The cotangent space of the irrelevant ideal and polynomial presentations

Let A be an ℕ-graded commutative R-algebra with irrelevant ideal A₊, the elements with zero constant term. By the graded Nakayama lemma TauCeti.GradedAlgebra.adjoin_eq_top_of_irrelevant_le, homogeneous elements of positive degree that generate A₊ modulo A₊ ^ 2 generate A as an algebra over its degree-zero part, so every R-subalgebra containing them and the degree-zero part is everything. Consequently, when the degree-zero part consists of scalars, homogeneous lifts of a spanning family of the cotangent space A₊ ⧸ A₊ ^ 2 generate A as an R-algebra, and the evaluation map from the polynomial ring on these lifts is surjective. Unlike the local Nakayama lemma, no finiteness hypothesis is needed: an induction on the degree replaces it.

Over a field k, the cotangent space thus bounds the Krull dimension: if it has finite dimension n, then dim A ≤ n. When equality holds, A is a polynomial algebra on n homogeneous generators of positive degree. Indeed, the evaluation map from a polynomial ring in n variables is then a surjection onto a ring of the same dimension, and a nonzero kernel would contain a nonzerodivisor of the polynomial ring, lowering the dimension.

This is the last step of the commutative-algebra proof that the invariants of a finite group generated by pseudoreflections form a polynomial algebra: once the invariant algebra is known to be regular at its irrelevant ideal, of dimension dim V, its cotangent space has dimension dim V, and the result here produces a basic system of homogeneous invariants.

Main results

• TauCeti.GradedAlgebra.span_image_toCotangent_irrelevant_eq_top: the cotangent space of A₊ is spanned by the images of homogeneous elements of positive degree.
• TauCeti.GradedAlgebra.bijective_aeval_of_irrelevant_le: over a field, Nat.card ι such generators, with Nat.card ι ≤ dim A, are algebraically independent and generate A.
• TauCeti.GradedAlgebra.ringKrullDim_le_finrank_cotangent: over a field, the dimension of A is at most the dimension of the cotangent space of A₊.
• TauCeti.GradedAlgebra.exists_bijective_aeval_of_finrank_cotangent_le: if moreover the dimension of the cotangent space is at most dim A, then A is a polynomial algebra on that many homogeneous generators of positive degree.

References

• W. Bruns, J. Herzog, Cohen–Macaulay Rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press (1993), § 1.5, for graded rings and the graded Nakayama lemma.
• R. Kane, Reflection Groups and Invariant Theory, CMS Books in Mathematics, Springer (2001), § 18, for its use in the polynomiality of invariants of reflection groups.

theorem TauCeti.GradedAlgebra.span_image_toCotangent_irrelevant_eq_top {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] : Submodule.span R (⇑(HomogeneousIdeal.irrelevant 𝒜).toIdeal.toCotangent '' {x : ↥(HomogeneousIdeal.irrelevant 𝒜).toIdeal | ∃ (n : ℕ), 0 < n ∧ ↑x ∈ 𝒜 n}) = ⊤

The cotangent space A₊ ⧸ A₊ ^ 2 of the irrelevant ideal is spanned over the coefficients by the images of the homogeneous elements of positive degree.

theorem TauCeti.GradedAlgebra.bijective_aeval_of_irrelevant_le {k : Type u_1} {A : Type u_2} [Field k] [CommRing A] [Algebra k A] (𝒜 : ℕ → Submodule k A) [GradedAlgebra 𝒜] (h𝒜 : ∀ x ∈ 𝒜 0, x ∈ ⊥) {ι : Type u_3} [Finite ι] {f : ι → A} (hf : ∀ (i : ι), ∃ (n : ℕ), 0 < n ∧ f i ∈ 𝒜 n) (h : (HomogeneousIdeal.irrelevant 𝒜).toIdeal ≤ Ideal.span (Set.range f) ⊔ (HomogeneousIdeal.irrelevant 𝒜).toIdeal ^ 2) (hdim : ↑(Nat.card ι) ≤ ringKrullDim A) : Function.Bijective ⇑(MvPolynomial.aeval f)

Polynomial presentation of a connected graded algebra. Let the degree-zero part of A consist of scalars, and let f be a finite family of homogeneous elements of positive degree generating the irrelevant ideal A₊ modulo A₊ ^ 2. If the number of elements is at most the Krull dimension of A, then f is algebraically independent and generates A: evaluation at f is an isomorphism from the polynomial ring.

theorem TauCeti.GradedAlgebra.ringKrullDim_le_finrank_cotangent {k : Type u_1} {A : Type u_2} [Field k] [CommRing A] [Algebra k A] (𝒜 : ℕ → Submodule k A) [GradedAlgebra 𝒜] [Module.Finite k (HomogeneousIdeal.irrelevant 𝒜).toIdeal.Cotangent] (h𝒜 : ∀ x ∈ 𝒜 0, x ∈ ⊥) : ringKrullDim A ≤ ↑(Module.finrank k (HomogeneousIdeal.irrelevant 𝒜).toIdeal.Cotangent)

Over a field, the Krull dimension of a connected graded algebra is at most the dimension of the cotangent space A₊ ⧸ A₊ ^ 2 of its irrelevant ideal, when the latter is finite.

theorem TauCeti.GradedAlgebra.exists_bijective_aeval_of_finrank_cotangent_le {k : Type u_1} {A : Type u_2} [Field k] [CommRing A] [Algebra k A] (𝒜 : ℕ → Submodule k A) [GradedAlgebra 𝒜] [Module.Finite k (HomogeneousIdeal.irrelevant 𝒜).toIdeal.Cotangent] (h𝒜 : ∀ x ∈ 𝒜 0, x ∈ ⊥) (hdim : ↑(Module.finrank k (HomogeneousIdeal.irrelevant 𝒜).toIdeal.Cotangent) ≤ ringKrullDim A) : ∃ (f : Fin (Module.finrank k (HomogeneousIdeal.irrelevant 𝒜).toIdeal.Cotangent) → A), (∀ (i : Fin (Module.finrank k (HomogeneousIdeal.irrelevant 𝒜).toIdeal.Cotangent)), ∃ (n : ℕ), 0 < n ∧ f i ∈ 𝒜 n) ∧ Function.Bijective ⇑(MvPolynomial.aeval f)

A connected graded algebra with cotangent space of the right dimension is polynomial. Let A be a graded algebra over a field k whose degree-zero part consists of scalars, and suppose that the cotangent space A₊ ⧸ A₊ ^ 2 of the irrelevant ideal has finite dimension n ≤ dim A. Then A is a polynomial algebra on n homogeneous generators of positive degree.