The exponent of Hironaka's J(m) on a chart #
For an affine open U of X, generators s of I(U) and a chart c ∈ s, the ideal of
J' · E^m on the chart of c is J'_c · (c)^m; for m large every generator g = y/cⁿ of J'_c
gives g · c^m = c^{m-n} y, a function on U that lies in I(U)^m and whose pullback lies in
J' · E^m on every chart a ∈ s — because (c/a)^{m-n} · (y/aⁿ) ∈ J'_a once m - n exceeds the
exponent of Hironaka.Scheme.BlowUp.Composite.Transition — hence lies in Hironaka's K_m(U). So
J'_c · (c)^m ≤ K_m(U) · Γ(X, U)[I(U)/c] for m ≥ m₀(U, c) (exists_chartIdeal_mul_pow_le), the
chart-level form of J' · E^m ≤ K_m.comap b; the exponents are finite maxima because X is
Noetherian (finitely generated I(U) and J'_c). This is the chart computation behind the
"sufficiently large integer m" of Hironaka's J(m) [Hir64, Ch. 0, §3, p. 133];
Hironaka.Scheme.BlowUp.Composite.Main takes the maximum over finitely many U and c.
(y/aⁿ) · a^{n+j} = a^j y in R[I/a].
c = a · (c/a) in R[I/a].
c^k y = a^{n+k} · ((c/a)^k · (y/aⁿ)) in R[I/a], for y ∈ Iⁿ.
From the exponent N of Hironaka.Scheme.BlowUp.Composite.Transition to all k ≥ N.
The chart ideal of J' · E^m is J'_a · (a)^m.
The exponent for a generator y/cⁿ of J'_c, uniform over the finitely many charts a ∈ s:
(c/a)^k · (y/aⁿ) ∈ J'_a for all k ≥ N and a ∈ s.
The membership c^k y ∈ K_{n+k}(U): for k ≥ N (the exponent of y/cⁿ), the function
c^k y ∈ I(U)^{n+k} lies in Hironaka's K_{n+k}(U) — on every chart a ∈ s its pullback is
a^{n+k} · ((c/a)^k · (y/aⁿ)) ∈ J'_a · (a)^{n+k}.
The generator step: a generator y/cⁿ of J'_c times c^{n+N+k} is the pullback of the
function c^{N+k} y ∈ K_{n+N+k}(U).
The generator step with the exponent bound: for a generator g = y/cⁿ of J'_c with exponent
N and m ≥ n + N, g · c^m is the pullback of a function of K_m(U).
The chart-level inequality: on the chart of c ∈ s, for m ≥ m₀(U, c),
J'_c · (c)^m ≤ K_m(U) · Γ(X, U)[I(U)/c].