Chart transitions of the ideals of J' and the exponent of Hironaka's J(m) #
The cross-chart step in the proof that K_m.comap b = J' · E^m for m large
(Hironaka.Scheme.BlowUp.Composite.Main). A generator g = y/a^n (y ∈ I^n) of the ideal J'_a
of J' on the chart of a is, on the chart of c, the fraction y/c^n times the unit
(a/c)^{-n} of the overlap D(a/c); since J'_a and J'_c cut out the same ideal sheaf on the
overlap, y/c^n lies in J'_c after multiplication by a power of a/c: there is N with
(a/c)^N · (y/c^n) ∈ J'_c.
- Ring level (
Rany commutative ring,a, c ∈ I):fracPow I n hy = y/a^n ∈ R[I/a];transitionHom_fracPow: the transitionR[I/a] → R[I/a]_{c/a} ≃ R[I/c]_{a/c}ofHironaka.Scheme.BlowUp.Transitionsendsy/a^nto(y/c^n)·(a/c)^{-n}— proved through the common embedding intoR_{ca};exists_pow_frac_mul_mem: from an identityP·R[I/c]_{a/c} = Q·R[I/c]_{a/c}of the transported ideals, the exponent for the elementsy/a^n ∈ P. - Scheme level: the two inclusions of the overlap
Spec R[I/c]_{a/c}into the blow-up — through the chart ofcand, via the transition, through the chart ofa— are equal (blowUp.hom_ext: both are admissible morphisms overX), so the inverse images ofJ'agree:chartIdeal_map_transition, the identityJ'_a·R[I/c]_{a/c} = J'_c·R[I/c]_{a/c}.
The charts of an affine blow-up glue along the localizations R_{g_j}, R_{g_l} ⊆ R_{g_j g_l}
[Hau14, Definition 4.13]. The overlap computations are also used for the stalks along the
exceptional divisor (Hironaka.Scheme.IdealSheaf.Order.ExceptionalOrder,
Hironaka.Scheme.Snc.ChartStalk).
The fraction y/aⁿ ∈ R[I/a] for y ∈ Iⁿ.
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The image of y/aⁿ in R_{c'} (a ∣ c') times aⁿ is y.
The transition R[I/a] → R[I/a]_{c/a} ≃ R[I/c]_{a/c} (Hironaka.Scheme.BlowUp.Transition) as
an R-algebra map into the overlap ring of the charts of a and c, in the coordinates of the
chart of c.
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- One or more equations did not get rendered due to their size.
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The transition on the fraction y/aⁿ: it is (y/cⁿ) · (a/c)⁻ⁿ, i.e.
τ(y/aⁿ) · (a/c)ⁿ = y/cⁿ. Checked after the injective embedding into R_{ca}, where both sides
are y · c⁻ⁿ.
The exponent at the ring level: if the ideals P ⊆ R[I/a] and Q ⊆ R[I/c]
generate the same ideal of the overlap ring R[I/c]_{a/c} (P transported by the transition), then
every y/aⁿ ∈ P satisfies (a/c)^N · (y/cⁿ) ∈ Q for some N.
I extends to the principal ideal (c) of the overlap ring R[I/c]_{a/c} (as on the chart
of c, then localized).
c stays a nonzerodivisor in the overlap ring R[I/c]_{a/c} (as in R[I/c], then
localization).
The inverse image along Spec ψ ≫ fromSpec_U of an ideal sheaf K on X is the ideal sheaf
of K(U)·S (through comap_fromSpec and comap_ofIdealTop_Spec_map).
The overlap D(a/c) of the charts of a and c, as the localization of the chart ring of
c at a/c.
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A morphism Spec T ⟶ blowUp I through the chart of b composed with the blow-up map is the
morphism Spec T ⟶ Spec Γ(X, U) ⟶ X of the composite ring map.
The overlap Spec R[I/c]_{a/c} ⟶ X is admissible for I: I extends to the principal ideal
generated by the nonzerodivisor c.
The two inclusions of the overlap Spec R[I/c]_{a/c} into the blow-up — through the chart of
c, and through the chart of a via the transition — agree: both are admissible morphisms over
X (blowUp.hom_ext).
The chart ideals of J' on the charts of a and c generate the same ideal of the overlap
ring R[I/c]_{a/c}, the ideal of a transported by the transition: each localizes to the ideal
of J' on the overlap D(a/c).