The stalks of the blow-up of a chart, read in the chart rings #
Hironaka.Scheme.Smooth.ChartIso identifies the blow-up B_{Z∩U} U of a chart U of étale
coordinates adapted to Z with the affine blow-up Proj of the Rees algebra of J = (v_0, …, v_{r-1}) ⊆ Γ(U), covers it by the standard charts Spec Γ(U)[J/v_j], and reads the stalk at a
point p of the chart of v_j as the localization Γ(U)[J/v_j]_p (Hau14, Definitions 4.15–4.16,
Theorem 4.19; [Kol07, Definition 60]). This module packages what the chart computation of the total
transform needs: at every point b of B_{Z∩U} U there are a chart index j < r, a prime p of
the chart ring and ring isomorphisms 𝒪_{B,b} ≃ 𝒪_{Proj, chart p} ≃ Γ(U)[J/v_j]_p under which the
stalk of the inverse image of any ideal sheaf D of X with D(U) = Q is Q · Γ(U)[J/v_j]_p
(exists_chart_point_map_stalkIdeal_eq). For D = Z this is the exceptional ideal (v_j); for a
component D = (v_c = 0) of the divisor it is the principal ideal of the image of v_c, which
Hironaka.Scheme.Snc.ChartSncData writes in the chart coordinates.
Why the lemmas hold. On the affine U ≅ Spec Γ(U) an ideal sheaf D with D(U) = Q is the
ideal sheaf of Q, so π_U^{-1} D corresponds to π^{-1}(specIdealSheaf Q) on the affine blow-up
(the isomorphism lies over U ≅ Spec Γ(U)). On the chart of v_j,
chart ≫ π = Spec (Γ(U) → Γ(U)[J/v_j]), so the pull-back of specIdealSheaf Q is
specIdealSheaf (Q Γ(U)[J/v_j]), whose stalk at p is Q Γ(U)[J/v_j] extended to the stalk of
Spec at p; the stalk of Spec at p is the localization at p, compatibly with the germs of
elements of the chart ring (exists_ringEquiv_stalk_localization). Open immersions and isomorphisms
induce isomorphisms of stalks carrying the stalk ideals of inverse images to the stalk ideals.
An ideal sheaf D of X with D(U) = Q on the affine open U is, on U ≅ Spec Γ(U), the
ideal sheaf of Q (the generalization of comap_toSpecΓ_specIdealSheaf from the centre to any
ideal sheaf).
The stalk of the ideal sheaf of Q' on Spec S at p is Q' extended along the germ map
(stalkIdeal_ofIdealTop in the specGerm spelling).
On the chart of a, the inverse image of the ideal sheaf of Q ⊆ R is the ideal sheaf of
Q R[J/a].
Under the isomorphism e of the stalk of the affine blow-up at the point chart p with the
localization R[J/a]_p, the stalk of the inverse image of the ideal sheaf of Q ⊆ R is
Q R[J/a]_p.
Every point of the affine blow-up of J = (v_0, …, v_{r-1}) lies in the chart of some v_j,
j < r.
The chart model of a point ([Hau14, Definitions 4.15–4.16, Theorem 4.19];
[Kol07, Definition 60]): every point b of B_{Z∩U} U lies in the chart of some v_j (j < r) as
a prime p of the chart ring Γ(U)[J/v_j], with ring isomorphisms
𝒪_{B,b} ≃ 𝒪_{Proj, chart p} ≃ Γ(U)[J/v_j]_p under which the stalk of the inverse image of any
ideal sheaf D of X with D(U) = Q is Q Γ(U)[J/v_j]_p.