The birational transform is the weak transform iff the mark is the order #
For a nowhere-zero ideal sheaf I and a nonempty connected centre Y, the birational transform
π_*^{-1}(I, m) = I_F^{-m} π⁻¹(I) (m ≤ ν_Y(I)) coincides with Hironaka's weak transform
I_F^{-ν} π⁻¹(I) exactly when m = ν_Y(I) [Kol07, Definition 48]. The direction (←) is the
uniqueness of the colon specification (Hironaka.Manifold.BlowUp.Transform.Colon) once the
exponents agree stalkwise: over the exceptional divisor the weak exponent is the generic order
along the component through the point, which is the order ν at every point of the connected
centre (Hironaka.Manifold.BlowUp.Transform.OrderAlong); off F the stalk of I_F is the
unit ideal and every colon is π⁻¹(I) itself.
The direction (→) is where the hypotheses I ≠ 0 and m ≤ ν are used. Krull's intersection
theorem in the Noetherian local stalk makes ν finite (ordAlong_ne_top). If m < ν, pick a'
over a in a blow-up chart with scaling coordinate u (the stalk of I_F is (u); the centre
has positive codimension, otherwise I_{Y,a} = 0 and ν = 0). Equality of the transforms says
(π⁻¹(I)_{a'} : u^m) = (π⁻¹(I)_{a'} : u^ν); since π⁻¹(I)_{a'} ⊆ (u^ν), every element of this
colon is u^{ν-m} times another one, so the colon lies in 𝔪 times itself and vanishes by
Nakayama, whence π⁻¹(I)_{a'} = 0 (Ideal.eq_bot_of_colon_pow_eq, in the domain stalk). Finally
the germ map 𝒪_{M, π a'} → 𝒪_{M', a'} of a blowing-up is injective
(IsBlowUp.germMap_injective): a germ vanishing near a' vanishes on the open image of a
neighbourhood of a point off F (where π is a local diffeomorphism; such points are dense),
and the identity theorem on a chart ball of M finishes; so I_a = 0, a contradiction. The
source states the coincidence without proof.
The algebraic core of the comparison of the
marked and the weak transform. In a Noetherian local domain, if J ⊆ (u^k), u ∈ 𝔪, and the
colons (J : u^m) and (J : u^k) agree for some m < k, then J = 0: every x ∈ (J : u^k)
satisfies x u^m ∈ J ⊆ (u^k), so x = u^{k-m} y with y ∈ (J : u^k); hence
(J : u^k) ⊆ 𝔪 (J : u^k) vanishes by Nakayama, and J = u^k (J : u^k) = 0.
The order of I
along Y at a ∈ Y is finite when I_a ≠ 0 — Krull's intersection theorem in the Noetherian local
stalk: ⋂_p I_{Y,a}^p = 0, so I_a ⊆ I_{Y,a}^p for every p would force I_a = 0.
The germ map of a blowing-up is
injective. A germ at π a' whose pull-back vanishes near a' vanishes on the open image of a
neighbourhood of a point b off the exceptional divisor (π is a local diffeomorphism
there, and such points are dense), hence — by the identity theorem on a chart ball of M around
π a' containing π b — near π a'.
The one-step form at exact order: if the order
of I along the centre is m at every point of Y, the marked transform of (I, m) is the weak
transform — the two colon specifications agree stalkwise (over Y the weak exponent is the generic
order, which is m; off the exceptional divisor both colons are by the unit ideal) and the colon
stalks determine the sheaf. This is the converse half of
birationalTransform_eq_weakTransformOf_iff below, in the form used along a succession of
blowings-up.
On a nonempty connected centre and for nowhere-zero I,
the birational transform of (I, m) is the weak transform iff m is the (generic) order ν
of I along Y [Kol07, Definition 48]. (←) The two colon specifications agree stalkwise — over
Y the weak exponent is the generic order, equal to ν = m by the constancy of the order along
the connected centre (ordAlong_eq_of_isPreconnected), and off F both colons are by the unit
ideal — and the colon stalks determine the sheaf. (→) If m < ν (finite by Krull,
ordAlong_ne_top), take a' over a (the centre has positive codimension, else I_{Y,a} = 0 and
ν = 0) in a blow-up chart with scaling coordinate u: the colons (π⁻¹(I)_{a'} : u^m) and
(π⁻¹(I)_{a'} : u^ν) agree, so π⁻¹(I)_{a'} = 0 by Nakayama in the domain stalk
(Ideal.eq_bot_of_colon_pow_eq), hence I_a = 0 by the injectivity of the germ map — against I ≠ 0.