Over ℂ, reduction commutes with the strict transform #
Bierstone–Milman remark that "it is easy to see that (X')_red = (X_red)'" [BM97, Remark 3.14];
this file proves it over ℂ. Write K = π^*I for the total transform, F = I_F for the
exceptional ideal (locally principal) and sat J = ⋃_k (J : F^k) for the saturation
(saturationStalk).
- The elementary half (any field):
sat(π^*√I) ⊆ √(sat π^*I), sinceπ^*√I ⊆ √(π^*I)(Ideal.map_radical_le), andg y^k ∈ √Kgivesg^n ∈ (K : y^{kn}) ⊆ sat K(iSup_colon_pow_le_radical_of_le_radical, with the principal generatoryofF_{a'},IsBlowUp.exists_span_singleton_stalkIdeal_exceptional). - The content (Rückert,
ℂ):sat Kis radical wheneverKis radical offF(mem_iSup_colon_pow_of_pow_mem). Letg^n ∈ sat Kata', sog^n ∈ (K : F^j)_{a'}; by coherence of the colon sheaf(K : F^j)(hasLocalGenerators_colon_idealSheaf) a representativesofgon an openW ∋ a'hass^n ∈ (K : F^j)(W), so offF(whereF^j = 𝒪)s^n ∈ Kand,Kbeing radical there,s ∈ K. On the open manifoldWas an analytic space, the colonC = (K|_W : ⟨s⟩)of the pulled-backKby the principal ideal sheaf ofsis coherent with|C| ⊆ |F| ∩ W; Rückert's Nullstellensatz puts every germ ofFata'into√C_{a'}, henceF_{a'}^m ⊆ C_{a'}for somem(the stalk is Noetherian), i.e.F^m · g ⊆ Kata', sog ∈ (K : F^m) ⊆ sat K. The stalk isomorphism of the inclusionW ⊆ M'(germMap_val_bijective) transports betweenM'andW. - Consequences:
isRadical_saturationStalk_of_isReduced(Xreduced impliesX'reduced, the form used for the geometric strict transform),saturationStalk_radical(the stalkwise identity√(sat π^*I) = sat(π^*√I)with the finite type of√Ias an explicit witness), and the form onIdealSheaf.radicalSubspace,strictTransformSubspace_radicalSubspace_of_hasLocalGenerators, given the four finite-type witnesses (Cartan's theorem forIand forX', the finite type of the two saturations).
The coherence of colons is Hironaka.AnalyticSpace.CoherentColon; Rückert's Nullstellensatz for
stalks, Hironaka.AnalyticSpace.Nullstellensatz.
Elementary algebra of saturations #
g y^k ∈ √K gives g ∈ √(sat_y K): the saturation of K' by y lies in the radical of the
saturation of K whenever K' ⊆ √K.
Every point: the stalk of the exceptional ideal sheaf is principal (a coordinate germ of a blow-up chart on the divisor, the unit ideal off it).
The Rückert core over ℂ #
The stalkwise colon of two ideal sheaves on a complex manifold has local generators (at the
analytic space Sp(M')).
A germ in a stalk of an ideal sheaf is the germ of a section of the ideal sheaf on a
neighbourhood: for g^n ∈ K'_{a'} and s a representative of g, s^n is a section of K' near
a'.
The principal ideal sheaf generated by a section t over an open set V containing every
point: the stalk family (t_x) has local generators (t itself).
The Rückert core over ℂ: if K is radical off |F|, its saturation by F is radical —
a germ whose m-th power lies in ⋃_k (K : F^k) lies there itself.
Reduction commutes with the strict transform, over ℂ #
Over ℂ [BM97, Remark 3.14], the strict transform of
a reduced subspace is reduced — for I with radical stalks the saturation stalks are radical.
Over ℂ ([BM97, Remark 3.14]), stalkwise with the finite type of
√I as an explicit witness: √(sat π^*I) = sat(π^*√I).
Over ℂ, on strictTransformSubspace (unconditional — in the
degenerate branch the stalks are ⊤): the strict transform of a reduced subspace has radical
stalks.
Over ℂ ([BM97, Remark 3.14], "(X')_red = (X_red)'"), given the four
witnesses (Cartan's theorem for I and for X', the Noether lemma for the two saturations): the
form on IdealSheaf.radicalSubspace.