Local coordinates and minimal presentations #
Let Q be a local ring whose maximal ideal 𝔪 is generated by coordinates x : ι → Q, with
dual derivations Dᵢ xⱼ = δᵢⱼ. The gradient of a ∈ Q is the residue of (D₁ a, …, Dₙ a).
By the first-order Taylor expansion, an element a ∈ 𝔪 lies in 𝔪² exactly when its gradient
vanishes, so on 𝔪 / 𝔪² the gradient is an isomorphism onto κⁿ. This file uses it to change
coordinates and to put a quotient of Q in minimal form.
Over a ring K, say that x is a coordinate system of the local K-algebra Q if x generates
𝔪 and the differentials dxᵢ form a basis of Ω[Q⁄K]; the dual derivations are then the
coordinate functions of this basis. This is the setting of
TauCeti.mem_maximalIdeal_sq_and_isUnit_det_hessian_of_formallyUnramified, for instance the local
ring at a closed point of a smooth variety with separable residue field.
Coordinate systems exist whenever Ω[Q⁄K] is free of finite rank, 𝔪 is finitely generated and
the residue field is formally étale over K, for instance finite separable over a field K: by
the conormal sequence, 𝔪 / 𝔪² is then the fibre of Ω[Q⁄K] at the closed point, and lifts to
𝔪 of the basis of this fibre are coordinates.
For a local quotient A = Q ⧸ (g₁, …, g_m) with e = dim_κ 𝔪_A / 𝔪_A², some n - e of the
relations g_s together with e further elements generate 𝔪. Dividing out by those n - e
relations gives a local ring P with a coordinate system of e elements, and the remaining
relations lie in 𝔪_P². This is a minimal presentation of A. When m = n - e + 1,
as for the local ring at a point of a complete intersection curve in affine n-space with
e = 2, a single relation f remains and A ≅ P ⧸ (f) is a plane curve singularity, to which
TauCeti.mem_maximalIdeal_sq_and_isUnit_det_hessian_of_formallyUnramified applies.
Main results #
TauCeti.maximalIdeal_eq_span_range_of_span_eq_top: elements of𝔪whose gradients spanκⁿgenerate𝔪.TauCeti.span_residue_apply_eq_top_of_le_span_sup_sq: conversely, elements of𝔪that span𝔪modulo𝔪²have gradients spanningκⁿ.TauCeti.card_le_card_of_le_span_sup_sq: elements of𝔪that span𝔪modulo𝔪²are at least as many as the coordinates.TauCeti.linearIndependent_residue_apply_sum_elim: completing elements with independent gradients by elements independent modulo them and𝔪²keeps the gradients independent.TauCeti.exists_coordinates: a coordinate system exists whenΩ[Q⁄K]has a finite basis,𝔪is finitely generated and the residue field is formally étale overK.TauCeti.exists_minimal_presentation: the minimal presentation of a quotient ofQ.TauCeti.exists_hypersurface_presentation: a quotient ofQbyn - e + 1relations is a hypersurface in a local ring withecoordinates.
References #
- Stacks Project, Tag 00RU: the conormal sequence.
- Stacks Project, Tag 031I: the conormal sequence of a formally smooth quotient is split exact on the left.
- Stacks Project, Section 53.19, Tag 0C46: nodes of curves, plane curve singularities defined by a nondegenerate quadratic form.
The gradient of a * b with b ∈ 𝔪 is the residue of a times the gradient of b.
The gradient of a combination ∑ cₖ wₖ of elements wₖ ∈ 𝔪 is the combination of their
gradients with coefficients the residues of the cₖ.
The gradient vanishes on 𝔪².
If the gradient of a ∈ 𝔪 is a combination of the gradients of elements wₖ ∈ 𝔪, then a
is a combination of the wₖ modulo 𝔪².
The gradient of an element of the ideal generated by wₖ ∈ 𝔪 is a combination of the
gradients of the wₖ, a partial converse of TauCeti.mem_span_range_sup_maximalIdeal_sq.
Nakayama's lemma in coordinates. Elements of 𝔪 whose gradients span κⁿ generate
𝔪.
Elements of 𝔪 that span 𝔪 modulo 𝔪² have gradients spanning κⁿ, the converse of
TauCeti.maximalIdeal_eq_span_range_of_span_eq_top.
If elements u and w of 𝔪 together span 𝔪 modulo 𝔪², then the gradients of w and of
any family v span κⁿ as soon as the gradients of v span those of all of u.
Elements of 𝔪 that span 𝔪 modulo 𝔪² are at least as many as the coordinates: the dual
derivations make the classes of the coordinates xᵢ linearly independent in the cotangent space
𝔪/𝔪², whose dimension is therefore at least Fintype.card ι.
Let wₖ ∈ 𝔪 have linearly independent gradients, and let yⱼ ∈ 𝔪 be linearly independent
modulo (wₖ) + 𝔪²: a combination ∑ cⱼ yⱼ lies in (wₖ) + 𝔪² only if every cⱼ ∈ 𝔪. Then the
gradients of the wₖ and the yⱼ together are linearly independent.
Existence of coordinates. Let Q be a local K-algebra whose maximal ideal 𝔪 is finitely
generated, whose differentials Ω[Q⁄K] have a basis indexed by ι, and whose residue field is
formally étale over K. Then there are x : ι → Q generating 𝔪 whose differentials dxᵢ form a
basis of Ω[Q⁄K].
Minimal presentation of a quotient of a local ring with coordinates. Let the
differentials dxᵢ of generators x : ι → Q of 𝔪 form a basis of Ω[Q⁄K], and let
A = Q ⧸ (g_s : s ∈ σ) be a quotient with cotangent space 𝔪_A / 𝔪_A² of dimension e.
Then there are n - e of the relations, indexed by T ⊆ σ, and e elements y : Fin e → Q,
which together generate 𝔪, such that P = Q ⧸ (g_s : s ∈ T) is a local ring with the
coordinate system y: the images of the yⱼ generate 𝔪_P and their differentials form a basis
of Ω[P⁄K]. All relations lie in 𝔪_P², so A is the quotient of P by relations in 𝔪_P².
A quotient by n - e + 1 relations is a hypersurface. In the setting of
TauCeti.exists_minimal_presentation, suppose that A = Q ⧸ (g_s : s ∈ σ) has a cotangent
space of dimension e and that there are n - e + 1 relations. Then all relations but one,
g_{s₀}, cut out a local ring P = Q ⧸ (g_s : s ≠ s₀) with the coordinate system y : Fin e → Q,
the image f of g_{s₀} in P lies in 𝔪_P², and π induces P ⧸ (f) ≅ A. For a complete
intersection curve with a two-dimensional cotangent space, such as a node, this exhibits A as a
plane curve singularity.