Kollár's Proposition 94: ideals invariant under the automorphisms of the form 1 + B #
For R = K⟦x₁, …, xₙ⟧, B ⊂ 𝔪 an ideal and I any ideal, the following are equivalent
[Kol07, Proposition 94]: (1) I is invariant under every automorphism of the form 1 + B
(IsInvariantOnePlus I B, Hironaka/Algebra/Local/FormalAut.lean); (2) B · D(I) ⊂ I; (3)
Bʲ · Dʲ(I) ⊂ I for every j ≥ 1. Kollár's motivation: derivations are "essentially the first
order automorphisms", so an ideal should be invariant under a subgroup of automorphisms exactly
when it is invariant to first order; the setting being infinite-dimensional, he works out the
details.
The proof follows Kollár's: (3) ⇒ (1) by the Taylor expansion modulo 𝔪^{s+1}
(Hironaka/Algebra/Local/Taylor.lean) and Krull's intersection theorem; (1) ⇒ (2) by the
one-direction expansion along xᵢ ↦ xᵢ + λ b for s admissible values of λ (the shift
automorphisms of Hironaka/Algebra/Local/FormalAut.lean) and the invertibility of the Vandermonde
matrix (λᵢʲ); (2) ⇒ (3) by induction on j with the product rule. The derivative ideals are
those of the standard coordinate structure RegularCoords.stdMvPowerSeries on K⟦X⟧. The
proposition is applied, through the Cohen isomorphism, to the completed local rings in the proof
of the formal equivalence of hypersurfaces of maximal contact [Kol07, Theorem 92].
The product rule read backwards, and (2) ⇒ (3) for any coordinate ring #
J · D(K) ≤ D(J K) + K · D(J) (the product rule a ∂h = ∂(a h) - h ∂a, as in the last step
of the proof of [Kol07, Proposition 94]).
The implication (2) ⇒ (3) of [Kol07, Proposition 94]: if B · D(I) ≤ I then Bʲ · Dʲ(I) ≤ I
for every j ≥ 1, by induction with the product rule.
Proposition 94 on K⟦X⟧ #
∂^α f ∈ D^{|α|}(I) for f ∈ I (each Taylor term lies in D^{|α|}(I) B^{|α|}).
An automorphism g of the form 1 + B is the substitution Xᵢ ↦ Xᵢ + bᵢ with
bᵢ := g(Xᵢ) - Xᵢ ∈ B.
If Bʲ Dʲ(I) ≤ I for all j ≥ 1, bᵢ ∈ B and f ∈ I, then each Taylor term
(1/α!) (∂^α f) b^α lies in D^{|α|}(I) B^{|α|} ≤ I (the proof of [Kol07, Proposition 94]).
The implication (3) ⇒ (1) of [Kol07, Proposition 94]: if Bʲ Dʲ(I) ≤ I for all j ≥ 1, then
I is invariant under the automorphisms of the form 1 + B — g(f) ∈ I + 𝔪^{s+1} for every s
by the Taylor expansion and taylor_term_mem, and Krull's intersection theorem finishes.
The admissible, nonzero, pairwise distinct scalars λ₁, …, λ_N ∈ ℚ of Kollár's Vandermonde
argument ("use s different values", the proof of [Kol07, Proposition 94]): all of 1, …, N + 1
except the one excluded value.
If I is invariant under the automorphisms of the form 1 + B, b ∈ B, f ∈ I and λ is
admissible (exists_onePlus_shift), then ∑_{j=0}^{s} λʲ uⱼ ∈ I + 𝔪^{s+1} with
uⱼ := (bʲ/j!) ∂ᵢʲ f (the range form).
The first step of (1) ⇒ (2) in the proof of [Kol07, Proposition 94]: if I is invariant under
the automorphisms of the form 1 + B, b ∈ B, f ∈ I and λ is admissible, then
∑_{j=1}^{s} λʲ uⱼ ∈ I + 𝔪^{s+1} with uⱼ := (bʲ/j!) ∂ᵢʲ f (from the one-direction Taylor
formula and g_λ(f) ∈ I).
The second step of (1) ⇒ (2) in the proof of [Kol07, Proposition 94]: choosing s distinct
admissible λ₁, …, λ_s ∈ ℚ, the Vandermonde matrix (λᵢʲ) is invertible over ℚ, hence
u₁ = b ∂ᵢ f ∈ I + 𝔪^{s+1}.
The third step of (1) ⇒ (2) in the proof of [Kol07, Proposition 94]: by Krull's intersection
theorem, b ∂ᵢ f ∈ I for b ∈ B, f ∈ I and every coordinate i.
The implication (1) ⇒ (2) of [Kol07, Proposition 94]: since D(I) is generated by the f
and ∂ᵢ f (f ∈ I) and b f, b ∂ᵢ f ∈ I, B · D(I) ≤ I.
Kollár's Proposition 94, first form [Kol07, Proposition 94, (1) ⟺ (2)]: I is invariant
under the automorphisms of the form 1 + B iff B · D(I) ≤ I.
Kollár's Proposition 94, second form [Kol07, Proposition 94, (1) ⟺ (3)]: I is invariant
under the automorphisms of the form 1 + B iff Bʲ · Dʲ(I) ≤ I for every j ≥ 1.
I is MC-invariant with respect to m iff it is invariant under every automorphism of the
form 1 + MC_m(I) (for MC_m(I) ≤ 𝔪, e.g. ord I = m ≥ 1): [Kol07, Proposition 94] with
B = MC(I), read against the definition (53.1) of [Kol07, 53].