Canonical resolutions with invariants of a marked ideal

Let $(\mathcal{I},E,\mu )$ be a marked ideal on a smooth $K$-scheme $X$ of finite type (Marked ideals and their support). A canonical resolution of $(\mathcal{I},E,\mu )$ is a resolution $(X_i{)}_{0\le i\le m}$, $X_0=X$, together with functions $\mathrm{inv}\text{\hspace{0.17em}⁣}:\mathrm{supp}({\mathcal{I}}_i,E_i,\mu )\to {\mathbb{Q}}_{\ge 0}\times {\mathbb{Q}}_{\ge 0}^{\mathrm{\infty }},\nu \text{\hspace{0.17em}⁣}:\mathrm{supp}({\mathcal{I}}_i,E_i,\mu )\to {\mathbb{Q}}_{\ge 0},\rho \text{\hspace{0.17em}⁣}:\mathrm{supp}({\mathcal{I}}_i,E_i,\mu )\to \mathrm{Sub}(E_i),$ ${\mathbb{Q}}_{\ge 0}^{\mathrm{\infty }}$ being the lexicographically ordered set of infinite sequences in ${\mathbb{Q}}_{\ge 0}$ with finitely many nonzero entries, such that for every $i$: (i) the centers $C_i$ of the blow-ups are regular and are the locus where the pair $(\mathrm{inv},\rho )$ attains its maximum, in fact components of the maximal locus of $\mathrm{inv}$; (ii) the three invariants have finite ranges on each support; $\mathrm{inv}$ and the lexicographically ordered pairs $(\mathrm{inv},\nu )$ and $(\mathrm{inv},\rho )$ are upper semicontinuous there; equivalently, the auxiliary functions $\nu$ and $\rho$ are upper semicontinuous on every level stratum $\{\mathrm{inv}=a\}$; (iii) for $x\in \mathrm{supp}({\mathcal{I}}_{i+1},E_{i+1},\mu )$ with ${\sigma }_{i+1}(x)\in C_i$ one has $\mathrm{inv}(x)<\mathrm{inv}({\sigma }_{i+1}(x))$ or $\mathrm{inv}(x)=\mathrm{inv}({\sigma }_{i+1}(x))$ and $\nu (x)<\nu ({\sigma }_{i+1}(x))$, while for ${\sigma }_{i+1}(x)\mathrm{\notin }{C}_i$ one has $\mathrm{inv}(x)=\mathrm{inv}({\sigma }_{i+1}(x))$, $\nu (x)=\nu ({\sigma }_{i+1}(x))$ and $\rho (x)=\rho ({\sigma }_{i+1}(x))$; (iv) for every etale morphism $\phi \text{\hspace{0.17em}⁣}:X^{\mathrm{\prime }}\to X$ the induced sequence ${\phi }^{\ast }(X_i)$ is an extension of the canonical resolution of ${\phi }^{\ast }(\mathcal{I},E,\mu )$ and the invariants agree, $\mathrm{inv}({\phi }_i(x^{\mathrm{\prime }}))=\mathrm{inv}(x^{\mathrm{\prime }})$, $\nu ({\phi }_i(x^{\mathrm{\prime }}))=\nu (x^{\mathrm{\prime }})$ and $\rho ({\phi }_i(x^{\mathrm{\prime }}))=\rho (x^{\mathrm{\prime }})$. Order $\mathrm{Sub}(E_i)$ by writing the labels of a subset in increasing order in the fixed total boundary order, padding with zeros, and comparing the resulting sequences lexicographically, with $0$ below every divisor label. An empty subset is the all-zero sequence. This supplies the order used for relative upper semicontinuity of $\rho$ and maxima of $(\mathrm{inv},\rho )$; it does not introduce a new choice of boundary order.

For an arbitrary étale morphism $X^{\mathrm{\prime }}\to X$ (Étale morphism of schemes), the pullback input may have a non-quasi-compact source. In condition (iv), the same notion is extended to these étale-local marked-ideal data: on every finite-type open chart of $X^{\mathrm{\prime }}$ use the preceding definition, and require a finite global sequence with a finite-range invariant family whose restrictions agree with those chart constructions up to inserted isomorphisms. The finite-type assumption remains on the original input $X$; no existence assertion is made for unrelated non-quasi-compact inputs. The canonical-resolution proposition proves this extension for arbitrary étale pullbacks, with sequence length bounded by that of the original resolution. Empty inverse centers contribute only inserted isomorphism steps, and invariant equalities refer to the corresponding stages.

The invariant $\mathrm{inv}$ takes values in the lexicographic order; the resolution is canonical in the sense that it is uniquely determined by the invariants, which are intrinsic.