Canonical resolution under isomorphisms of the ground field

Pullback of the derived objects at their proper inputs. By [F1], derivative ideals of every coherent ideal commute with the semilinear isomorphism. For a maximal-order marked ideal $(\mathcal{J},\eta )$, finite sums, products and powers commute with this pullback, so [F2] gives ${\phi }^{\ast }H(\mathcal{J},\eta )=H({\phi }^{\ast }\mathcal{J},\eta )$ and the analogous identity for $C(H(\mathcal{J},\eta ))$. For a general input $(\mathcal{I},\mu )$ first transport its monomial and non-monomial factors: local-ring isomorphisms preserve divisibility by each ordered boundary equation and preserve the maximum residual order on the corresponding supports. If that maximum is positive, the corresponding companions $O(\mathcal{I},\mu )$ are maximal-order inputs, and it is these companions to which the homogenization and coefficient identities apply. If the maximum is zero, the inputs are monomial near their supports and Step 2b applies directly; no homogenization of an unrestricted input is used.

Pullback of orders, supports and strata. By [F3] one has ${\mathrm{ord}}_{\phi (x)}(\mathcal{J})={\mathrm{ord}}_x({\phi }^{\ast }\mathcal{J})$ for every coherent ideal $\mathcal{J}$ on $X$, hence $\mathrm{supp}({\phi }^{\ast }(\mathcal{J},\nu ))={\phi }^{-1}(\mathrm{supp}(\mathcal{J},\nu ))$, and $s_{{\phi }^{-1}E}(x)=s_E(\phi (x))$; since the SNC condition is stalk-local, $\phi$ maps the strata $H_{\alpha }^s$ of the algorithm for $(\mathcal{I},E,\mu )$ isomorphically onto the corresponding strata for ${\phi }^{\ast }(\mathcal{I},E,\mu )$, and carries the restriction $(\mathcal{I}{\mathrm{\mid }}_{H_{\alpha }^s},\mu )$ to $({\phi }^{\ast }\mathcal{I}{\mathrm{\mid }}_{{\phi }^{-1}{H}_{\alpha }^s},\mu )$.

Commutation with the reductions of the algorithm. We prove the assertion by induction on $d=\dim X$. For $d=0$, every component is the spectrum of a finite separable field extension, and the hypothesis of generic nonvanishing makes the ideal the unit ideal on each component. The positive marking therefore gives empty support and the canonical sequence is the identity; its base change is again the identity by step 2.1. Assume now $d\ge 1$ and the assertion known in dimensions $<d$, and transport the algorithm of [F6] along the identification of steps 1.1-2.1: (a) for every maximal-order input reached in Step 1, the replacement of $(\mathcal{J},\nu )$ by the equivalent $C(H(\mathcal{J},\nu ))$ commutes with pullback by step 1.1; (b) the strata $H_{\alpha }^s$ and the restricted coefficient ideals correspond by step 2.1. First remove components contained in the support by Step 1aa: their regular SNC stratum blowups and controlled division commute with the isomorphism, and both sides give the same count/infinity primary value with $\nu =0$, $\rho =\mathrm{\varnothing }$. No lower-dimensional induction is applied to these zero restrictions. By the coefficient-ideal support identity in [F6], the remaining restrictions are generically nonzero on each retained component; their lower-dimensional canonical resolutions therefore commute with $\phi$ by induction. Once the inherited boundary is disjoint from the support, the isolated codimension-one components of Step 1ba correspond by their local equations and labelled Cartier division. Only on the remaining codimension-at-least-two support do we pass to Step 1bb; (c) a hypersurface of maximal contact $V(u)$, $u\in T(\mathcal{J})$, is carried to the hypersurface $V({\phi }^{\ast }u)$ with ${\phi }^{\ast }u\in T({\phi }^{\ast }\mathcal{J})$ by steps 1.1-2.1, and the restriction of $C(\mathcal{J},\nu )$ to $V(u)$ corresponds, so the induction hypothesis applies to the lower-dimensional marked ideal on $V(u)$; (d) the monomial/non-monomial decomposition $\mathcal{J}=M(\mathcal{J})N(\mathcal{J})$ is read off from the vanishing orders of $\mathcal{J}$ along the members of $E$ by [F3], hence is transported, and, when the residual maximum is positive, its companion satisfies ${\phi }^{\ast }(O(\mathcal{J}))=O({\phi }^{\ast }\mathcal{J})$; when it is zero the local monomial Step 2b data correspond directly; (e) the invariants assembled in [F6] from $s_E$, the order functions and the lower-dimensional invariants have equal values at corresponding points by steps 1.1-2.1 and the induction hypothesis, and equality of all values together with preservation of their orders carries the maximal locus of $(\mathrm{inv},\rho )$ on $X_i$ to its exact preimage under ${\phi }_i$ on $X_i^{\mathrm{\prime }}$; its closedness is supplied by [F8]. Since the algorithm's centre at each stage is exactly that maximal locus [F6], the process for ${\phi }^{\ast }(\mathcal{I},E,\mu )$ has centres ${\phi }_i^{-1}(C_i)$ and produces the base-changed marked ideals $({\mathcal{I}}_i^{\mathrm{\prime }},E_i^{\mathrm{\prime }},\mu )$.

The canonical resolution of the pullback. By [F4, F7] the sequence $(X_i^{\mathrm{\prime }})=(X_i{\times }_X{X}^{\mathrm{\prime }})$ is a multiple test blow-up of ${\phi }^{\ast }(\mathcal{I},E,\mu )$ with ${\mathcal{I}}_i^{\mathrm{\prime }}={\phi }_i^{\ast }{\mathcal{I}}_i$, $E_i^{\mathrm{\prime }}={\phi }_i^{-1}{E}_i$, and each ${\phi }_i$ is an isomorphism; by step 3.1 its centre at each stage is ${\phi }_i^{-1}(C_i)$, which is the centre prescribed by the algorithm for the pullback, and the algorithm is determined by the intrinsic data [F6]. Hence $(X_i^{\mathrm{\prime }})$ is the canonical resolution of ${\phi }^{\ast }(\mathcal{I},E,\mu )$. For $x\in X_i^{\mathrm{\prime }}$ with ${\phi }_i(x)\in \mathrm{supp}({\mathcal{I}}_i,E_i,\mu )$ step 2.1 gives $x\in \mathrm{supp}({\mathcal{I}}_i^{\mathrm{\prime }},E_i^{\mathrm{\prime }},\mu )$, and step 3.1(e) gives $\mathrm{inv}({\phi }_i(x))=\mathrm{inv}(x)$ and $\nu ({\phi }_i(x))=\nu (x)$; since ${\phi }_i$ carries the ordered family $E_i^{\mathrm{\prime }}={\phi }_i^{-1}{E}_i$ isomorphically onto $E_i$, the subsets of exceptional divisors through corresponding points match and $\rho ({\phi }_i(x))={\phi }_i(\rho (x))$. This completes the induction and the proof.