Canonical principalization of ideals in characteristic zero

Principalization from the resolution. Let $(X_i{)}_{0\le i\le r}$ be the canonical resolution of $(\mathcal{I},\mathrm{\varnothing },1)$ from [F1] and let $({\mathcal{I}}_i,1)$ be the controlled transforms. Since $\mathrm{supp}({\mathcal{I}}_r,1)=\mathrm{\varnothing }$, the ideal ${\mathcal{I}}_r$ is the unit ideal by [F2]; unwinding the transform rule we get ${\sigma }_i^{\ast }{\mathcal{I}}_{i-1}=\mathcal{I}(D_i){\mathcal{I}}_i$ at each step, so the total transform ${\sigma }_r^{\ast }\mathcal{I}$ is a product of powers of the exceptional divisors $D_i$ and their strict transforms, each a component of $E_r$ by [F4]. Hence ${\sigma }_r^{\ast }\mathcal{I}$ is the ideal of an effective Cartier divisor $\tilde{E}$ with SNC support which is a natural combination of the irreducible components of $E_r$, and by [F4] both the exceptional divisors and the centers are in SNC position: clauses (a) and (b).

Intrinsic constants on a component. For an integral open-and-closed component $Z$ of $X$, put $R=\mathrm{\Gamma }(Z,{\mathcal{O}}_Z)$ and let $L$ be its elements algebraic over $K$. The minimal polynomial expresses the inverse of each nonzero such element as a polynomial in it, so $L$ is a field. On an affine chart $\mathrm{Spec}A\subseteq Z$, [F5] makes $A$ finite over a polynomial ring $K[z]$. Every finite subextension $L_0\mathrm{/}K$ gives a subfield $L_0(z)$ of $A{\otimes }_{K[z]}K(z)$, so $[L_0:K]$ is bounded by the module-generator count; choosing a maximal degree proves $L\mathrm{/}K$ finite, hence separable. Any field subring of $R$ must lie in $L$: if it contains $f$ transcendental over $K$, every $f-q$, $q\in \mathbb{Q}$, is a unit. By [F5], the nonzero finite-type $K(f)$-algebra $A{\otimes }_{K[f]}K(f)$ has a finite residue field $E\mathrm{/}K(f)$. Images $b_i$ of finitely many generators of $A$ become integral over $K[f,1\mathrm{/}p]$ after clearing finitely many denominators, with $p\mathrm{\ne }0$. The algebra $B=K[f,1\mathrm{/}p,b_1,\dots ,b_s]$ is integral over $K[f,1\mathrm{/}p]$ and receives a map from $A$. Choose $q\in \mathbb{Q}$ with $p(q)\mathrm{\ne }0$; lying over supplies a prime containing $f-q$ in $B$, contradicting its being the image of a unit of $A$. Thus $L$ is the unique largest field subring of $R$. Scheme automorphisms, including those permuting components, consequently induce isomorphisms of these intrinsic fields.

Canonicity and smooth naturality. The resolution, hence the principalization, is determined by its canonical invariant centers. The smooth and ambient-embedding comparisons in [F3] transport these centers and the controlled-transform rules, so they transport the total-transform factorization from step 1.1 as well. This proves canonicity and both commutation clauses.

Equivariance for ideal-preserving actions. Regard each component as an $L$-scheme. A smooth affine $L$-ambient presentation is smooth over $K$ because $L\mathrm{/}K$ is finite separable. Every $K$-derivation annihilates $L$ by its separable minimal polynomials, so the $K$- and $L$-derivative ideals agree; orders, boundary strata, homogenizations, coefficient ideals and companion ideals then agree, giving the same canonical sequence. An automorphism preserving $\mathcal{I}$ transports the componentwise marked ideals by a semilinear isomorphism of their intrinsic constant fields. By [F3] it therefore transports each canonical center, and lifts successively to the blowups. These lifts satisfy identity and composition: the blowup lifts induced from the ideal identifications are natural, and equivalently two lifts agree on the dense complement of the centers and hence on the reduced separated smooth resolution. Thus any ideal-preserving abstract group action lifts coherently, including actions not preserving $K$.