Canonical resolution of marked ideals

Induction setup. We argue by induction on $d=\dim X$, componentwise on the disjoint irreducible components of the smooth scheme. For $d=0$, $X$ is a finite disjoint union of $K$-points; the generic-nonzero hypothesis makes $\mathcal{I}={\mathcal{O}}_X$ on every component, so the support is empty and the trivial sequence is the canonical resolution. Assume the proposition for smooth schemes of dimension $<d$ and marked ideals satisfying the stated positive-marking and componentwise generic-nonvanishing hypotheses; by [F2] we may replace any marked ideal of maximal order by its equivalent $C(H(\mathcal{J},\mu ))$, and we do so without changing the supports or the admissible centers.

Step 1a, eliminating strata contained in the support. At the start of each nonterminal Step 1a pass at stage $i$, compute and hold fixed for that pass $s_i=\max \{s_i(x):x\in \mathrm{supp}({\mathcal{J}}_i,E_i,\mu )\}$, where $s_i(x)$ counts strict transforms of the boundary family $E$ fixed when this maximal-order Step 1 begins; new exceptional divisors created during Step 1a remain in the full $E_i$ but are not added to this boundary count. If the support is empty, terminate; if $s_i=0$, finish Step 1a and proceed to Step 1b without restricting to $H^0=X$. For $s_i>0$, use only irreducible components $H_{\alpha ,i}^{s_i}$ of intersections of exactly $s_i$ boundary components that meet the support. These strata are smooth and have finitely many pairwise disjoint irreducible components; taking them componentwise ensures that each retained restricted ideal is generically nonzero unless the stratum belongs to the contained-support branch. Their intersections with the support are pairwise disjoint: a point in two distinct such pieces would lie on more than $s_i$ boundary components. The ambient intersections can overlap away from the support, where the induced test blow-ups are identities. If $H_{\alpha ,i}^{s_i}\subseteq \mathrm{supp}({\mathcal{J}}_i,E_i,\mu )$, blow up this stratum. For $s_i\ge 2$, the boundary count drops at every support point above it; for $s_i=1$, [F5] shows that the controlled transform has empty support above the codimension-one stratum. Assign points of this stratum the source invariant $(s_i,\mathrm{\infty },0,0,\dots )$, represented by $((0,s_i),(1,0),(0,0),(0,0),\dots )$ under [A1], with $\nu =0$ and $\rho =\mathrm{\varnothing }$. This is lexicographically above the finite lower-dimensional invariant values on the other support pieces. Recompute $s$ only at the start of the next pass.

Step 1b, codimension-one components. After Step 1a reaches $s=0$, no strict transform of the inherited Step 1 boundary family meets the support. If the support has codimension-one components, [F5] shows they are smooth and isolated from the remaining support. They are strict transforms of the original such components: maximal order makes the ideal order at the generic point of each earlier center exactly its marking, so division by that exceptional power leaves no new exceptional component in the support. Every earlier center is contained in an isolated component or disjoint from it. For a center contained in a smooth component $C=(u=0)$, choose adapted regular parameters for that center including $u$ and the equations of the exceptional divisors already transverse to $C$; this is possible because the center is regular and SNC with that boundary. In a blowup chart indexed by another center parameter $w$, the strict transform of $C$ has equation $u\mathrm{/}w=0$ and the new exceptional divisor has equation $w=0$; the chart indexed by $u$ misses the strict transform. These coordinate charts preserve simultaneous SNC of $C$ with the newly created boundary members, by induction over the preceding blowups. Since the inherited boundary no longer meets the support, $C$ has SNC with the full current boundary. Its labelled Cartier blowup is therefore admissible under [F5]; assign them the source invariant $(0,\mathrm{\infty },0,\dots )$, represented under [A1] by $((0,0),(1,0),(0,0),\dots )$, with $\nu =0$ and $\rho =\mathrm{\varnothing }$. This is above all finite lower-dimensional invariant values, so these components are the next centers. Blowing them up makes the controlled-transform support empty above them. The remaining support has codimension at least two.

Step 2b, the monomial case. If the support is empty, terminate. Otherwise, for a monomial ideal $\mathcal{I}=M(\mathcal{I})={\prod }_D{x}_D^{a_D}$, set $\mathrm{inv}=(0,0,\dots )$ and $\nu (x)={\mathrm{ord}}_x(\mathcal{I})\mathrm{/}\mu$. At each point, let $A$ be the inclusion-minimal threshold subset selected by the source's maximal $\rho$ rule: ${\sum }_{D\in A}{a}_D\ge \mu$, while ${\sum }_{D\in A\setminus \{D_0\}}{a}_D<\mu$ for every $D_0\in A$. The maximal locus of $(\mathrm{inv},\rho )$ gives the regular SNC center locally defined by the divisors in this selected $A$. In a blow-up chart indexed by any $D_j\in A$, the new exceptional exponent is $a={\sum }_{D\in A}{a}_D-\mu <a_{D_j}$, by the minimality condition with $D_0=D_j$. For any surviving support point $q$ above $p\in C$, the old strict boundary divisors through $q$ are among those through $p$, except for the chart divisor $D_j$, whose old exponent is replaced by $a<a_{D_j}$. Hence ${\mathrm{ord}}_q(M_{\mathrm{n}\mathrm{e}\mathrm{w}})\le a+{\sum }_{D\ni p,D\mathrm{\ne }{D}_j}{a}_D<{\mathrm{ord}}_p(M)$; divisors missing $q$ only lower the sum. Thus at every support point above the center the total monomial order drops, so $\nu$ strictly drops while the monomial-case $\mathrm{inv}$ stays fixed; equivalently, $(\mathrm{inv},\nu )$ strictly descends. There are finitely many initial maximal support components. Each chosen component is replaced by finitely many maximal components above it, and the maximum monomial order on each new component is strictly smaller, as shown by the chart inequality. This is a finitely branching replacement tree whose path length is bounded by the initial integer monomial-order maximum. It is finite, so the process terminates with empty support; the discreteness of $\nu \in \frac{1}{\mu }{\mathbb{Z}}_{\ge 0}$ provides this integer bound.

Step 1ab, moving support away from maximal boundary strata. This branch occurs only for the pass-start value $s_i>0$, after all strata contained in the support have been blown up. Resolve each lower-dimensional restricted marked ideal on $H_{\alpha ,i}^{s_i}$. These restrictions are generically nonzero: at the generic point of a retained component the ambient support is absent, and [F4] identifies the restricted support with its ambient intersection. The restricted support pieces are disjoint, so their canonical sequences glue; on ambient overlaps outside the support there are no centers and the induced maps are identities. By [F4], each restricted support equals the corresponding ambient support intersection, and the glued sequence is a multiple test blow-up of the ambient ideal. On a support piece use $\mathrm{inv}(x)=((0,s_E(x)),{\mathrm{inv}}_{{\mathcal{J}}_i{\mathrm{\mid }}_{H_{\alpha ,i}^{s_i}}}(x))$, encoding the finite leading coordinate by [A1], with the restricted $\nu$ and $\rho$; the support piece determines the stratum uniquely. By induction, above each center the pair $(\mathrm{inv},\nu )$ strictly descends, while off the center the induced map is the identity and the invariants are unchanged. Once all these restrictions are resolved, no support point remains on any maximal stratum from the pass-start boundary count, so the maximum boundary count drops before the next pass; recompute it only at that next pass. As the count is a nonnegative integer, finitely many passes reach $s_i=0$, and then Step 1b begins.

Step 1b, hypersurfaces of maximal contact. When the inherited Step 1 boundary misses the support, first remove codimension-one support components as in 1.3. For the remaining codimension-at-least-two support, use the strict transforms of tangent hypersurfaces chosen on finitely many initial Step 1 charts, with the inherited boundary temporarily forgotten. These charts cover the surviving support: the original isolated codimension-one pieces have already been removed. By Derivative ideals under a multiple test blow-up, their controlled equations remain in the transformed tangent ideal, and The supports of a multiple test blow-up stay inside the strict transforms of a hypersurface of maximal contact puts the support in their strict transforms. They are simultaneously transverse to the newly created boundary: initially that boundary is empty; if a regular center lies in $V(u)$ and is SNC with the existing boundary transverse to $V(u)$, choose adapted center parameters including $u$. In a chart indexed by another center parameter $w$, the equations are $u\mathrm{/}w$ for the strict hypersurface and $w$ for the new exceptional divisor, along with the surviving old boundary parameters; the $u$-chart has no hypersurface support. Thus transversality persists. The inherited boundary now misses the support, so these transformed hypersurfaces are transverse to the full boundary there, as [F3] requires. Write $V(u)$ for such a transformed hypersurface. Its coefficient-ideal restriction is generically nonzero on each component, since [F4] identifies its support with a subset of ambient codimension at least two, hence a proper subset of every hypersurface component. By [F4], the inductive resolution on $V(u)$ transfers to the ambient ideal; off the restricted support, this transfer is an isomorphism. Set $\mathrm{inv}(x)=((0,0),{\mathrm{inv}}_{\mathcal{J}{\mathrm{\mid }}_{V(u)}}(x))$, with $\nu ,\rho$ from the restriction. Apply [F3] to two initial tangent directions with the inherited boundary forgotten, and lift its étale comparisons along the preceding test sequence. The new exceptional divisors are preserved by these lifted comparisons; near the surviving support the inherited boundary is absent. Hence they identify the pulled-back homogenized ideals and the full current boundary, so the values, centers, and local resolutions agree on overlaps. Thus above every transferred center $(\mathrm{inv},\nu )$ strictly descends by induction, and off the center all invariants are unchanged. The local sequences glue to the canonical resolution.

Step 2, the companion reduction and monomial case. If $\mathrm{supp}(\mathcal{I},E,\mu )=\mathrm{\varnothing }$, the resolution is complete. Otherwise factor $\mathcal{I}=M(\mathcal{I})N(\mathcal{I})$ and compute $r={\mathrm{ord}}_N(\mathcal{I})$ as in [F7]. If $r=0$, $N(I)$ is a unit on a neighbourhood of the support, so the marked ideal agrees there with $(M(I),\mu )$: resolve this neighbourhood by Step 2b and extend by the identity outside its support, without forming a companion. The maximal-order Step 1 may use $(M(\mathcal{I}),\mu )$ only when the original input is already of maximal order. If $r>0$, form the maximal-order companion $O(\mathcal{I},\mu )$ and apply Step 1. For $r<\mu$, a companion test sequence is simultaneously admissible for $(N,r)$ and $(M,\mu -r)$, and their product transforms to $(I_i,\mu )$ by the product rule. For $r\ge \mu$, the controlled transform of $I$ is the transformed $(N,r)$ times a monomial factor, with each new exponent containing $r-\mu \ge 0$. Thus every companion center is admissible for $(I,\mu )$ in either case. Write $N_i^{\ast }$ for the transform of $(N,r)$; it remains of maximal order by Controlled transforms preserve maximal order on nonempty transformed schemes. Factoring its exceptional powers gives $N(I_i)$, so $\mathrm{ord}N(I_i)\le \mathrm{ord}{N}_i^{\ast }\le r$. At a point with $\mathrm{ord}N(I_i)=r$, every exceptional factor removed from $N_i^{\ast }$ is a unit there, so $\mathrm{ord}{N}_i^{\ast }=r$. If that point is in the original marked support, it is in the transformed companion support by the sum-support rule (or directly when $r\ge \mu$). Consequently clearing the companion support lowers the maximum order of $N(I_i)$ on the remaining original support, so the first source coordinate $r\mathrm{/}\mu$, represented by $(0,r\mathrm{/}\mu )$ under [A1] in $\mathrm{inv}=((0,r\mathrm{/}\mu ),{\mathrm{inv}}_O)$ on the transformed companion support, strictly drops between companion passes. This formula is used only where ${\mathrm{inv}}_O$ is defined; values at other original-support points are assigned by the successive-transfer construction below. Within a companion pass, Step 1 gives strict descent of $({\mathrm{inv}}_O,{\nu }_O)$ above every center; therefore $(\mathrm{inv},\nu )$ strictly descends there, and off the center the induced map is the identity and all invariants are unchanged. Repeat only while $r>0$; the finite sequence of strictly decreasing nonnegative integer values of $r$ ends with empty support or $r=0$. In the latter case, Step 2b has constant monomial-case $\mathrm{inv}$ and strictly lowers $\nu ={\mathrm{ord}}_x(M(\mathcal{I}))\mathrm{/}\mu$ above every center, so $(\mathrm{inv},\nu )$ strictly descends until the support is empty.

Define the invariants on the entire support by successive assignment, rather than evaluating a companion invariant outside its domain. During a positive-$r$ pass assign the tuple in 3.1 only on the transformed companion support; within Step 1 assign boundary-stratum and maximal-contact values only on their stated domains. Each point not yet assigned lies off every center in that pass, so it has a unique unchanged lift through the pass, with the same local ideal and boundary germs. After that pass continue the algorithm on the remaining support. When its residual-order stratum, boundary-count stratum, or final monomial branch is reached, assign the corresponding invariant there and transfer it back through these unchanged lifts. Termination proved in the preceding steps guarantees a finite first assignment for every point still in a support: otherwise an unassigned point would survive to the empty final support. Use the same transferred value at all earlier stages where the point was untouched. In particular a point of residual order below $r$ receives its own later residual-order coordinate, not the current $r\mathrm{/}\mu$ and not an undefined ${\mathrm{inv}}_O$. The analogous convention fills lower boundary-count pieces during Step 1. Thus the active maximum is unchanged, and the invariants at points outside every intervening center are unchanged by construction.

Descent, termination, and canonicity. In a boundary pass the fixed pass-start count makes each maximal-stratum blow-up lower $s_E$ above its center; on restricted strata, induction strictly lowers $(\mathrm{inv},\nu )$ above centers, and the ambient maps are identities off the support intersections. The codimension-one branch assigns the top source value $(0,\mathrm{\infty },0,\dots )$, rationally encoded as $((0,0),(1,0),(0,0),\dots )$; blowing up that component makes the controlled-transform support empty above it. In the hypersurface branch, induction strictly lowers $(\mathrm{inv},\nu )$ above every transferred center and the maps are identities off the restricted support. In the companion branch, the first source coordinate $r\mathrm{/}\mu$ (encoded as $(0,r\mathrm{/}\mu )$) strictly decreases between passes and the companion invariant pair strictly decreases within a pass. In the monomial branch, $\mathrm{inv}$ is constant and $\nu$ strictly decreases above each center by 1.4. Thus in every branch the pair $(\mathrm{inv},\nu )$ strictly decreases above centers, while off centers the morphism is an isomorphism and all invariants are unchanged; each loop terminates by the decreasing boundary count, the decreasing integer $r$, induction in lower dimension, or the discrete nonnegative values of $\nu$. The values and maximal centers are determined by boundary counts, orders, the threshold-subset rule and lower-dimensional canonical invariants. Their semicontinuity is proved in the next steps.

Finite ranges and lexicographic assembly. Each stage has finitely many boundary components and finitely many maximal-contact charts. There are finitely many passes by step 5.1; in each dimension reduction the finite collection of restricted stage invariants has finite range by induction, and the tuple depth is uniformly bounded by the ambient dimension. Residual orders, boundary counts and the exponents of the finitely many monomial ideals occurring in these passes take finitely many values. Consequently all three invariants have finite ranges on each support; padding the bounded-depth rational tuples adds no values. For the finite-stratum assembly used below, let $f$ be a finite-range upper semicontinuous function on a support $S$. For each attained value $a$, its level stratum $S_a=\{f=a\}$ is locally closed, since $\{f>a\}$ is a finite union of closed superlevels. If $F=\{x\in S_a:g(x)\ge b\}$ is closed in $S_a$, then its closure in $S$ lies in $\{f\ge a\}$ and meets $S_a$ exactly in $F$. Thus $\bar{F}\subseteq F\cup \{f>a\}$, and the lexicographic superlevel $\{(f,g)\ge (a,b)\}=\{f>a\}\cup F=\{f>a\}\cup \bar{F}$ is closed. For unattained $a$ this superlevel is just $\{f>a\}$. Conversely, restricting pair superlevels to $S_a$ gives relative superlevels of $g$.

Primary and auxiliary semicontinuity. On each residual-order or boundary-count stratum, unchanged lifts in step 4.1 identify the local ideal and boundary germs with their later pass. The leading residual-order and boundary-count functions have closed superlevels by [F6]. On a fixed leading level, the terminal infinity branch is closed and has maximal tail; distinct maximal boundary pieces are disjoint on the support, and their lower-dimensional primary tails are upper semicontinuous by induction. In the nonboundary branch, the isolated codimension-one pieces have the maximal infinity tail; on the remainder, the finitely many maximal-contact charts identify their primary tails étale-locally by [F3], so relative closed superlevels glue. Repeated application of step 6.1, with the leading coordinate as $f$ and the lower-dimensional tail as $g$, proves upper semicontinuity of the entire primary invariant. Now fix its complete value $a$. In every recursive branch this fixes the leading coordinate and the entire lower-dimensional primary tail, so the inductive auxiliary semicontinuity on that lower primary level applies. Terminal infinity branches have constant $\nu =0$, $\rho =\mathrm{\varnothing }$. On a monomial branch, $\nu$ is a finite sum of nonnegative exponents of boundary divisors through the point divided by the mark, hence has closed superlevels. Each eligible inclusion-minimal threshold subset contributes its closed boundary intersection; the maximum subset rule makes each $\rho$ superlevel a finite union of these intersections. Boundary pieces are disjoint on the relevant support; maximal-contact charts agree by [F3]; unchanged lifts transport these relative-closed loci through earlier passes. Hence $\nu$ and $\rho$ are upper semicontinuous on $\{\mathrm{inv}=a\}$. Applying step 6.1 to the primary invariant and each auxiliary function proves upper semicontinuity of $(\mathrm{inv},\nu )$ and $(\mathrm{inv},\rho )$ on the entire support. In particular their finite ranges yield closed maximal loci; the branch constructions prove that the center selected by $(\mathrm{inv},\rho )$ is regular and has SNC with the boundary.

Étale canonicity. Under an étale morphism, the tracked Step 1 boundary components, orders, monomial and non-monomial factors, threshold subsets, companion supports and maximal-contact restrictions pull back compatibly. The glueing equality [F3] identifies maximal-contact choices, while [F4] and [F8] preserve restrictions, test sequences and equivalences. Thus the primary and both auxiliary values agree at corresponding points, the pair maxima give the pulled-back centers, and empty centers contribute only identity steps. The induced sequence is an extension of the canonical resolution with agreeing invariants. For an arbitrary étale source $X^{\mathrm{\prime }}$, take finite-type open charts: étale local finite presentation over the finite-type $X$ gives such charts, and the preceding finite-type comparison applies on each. Use the pulled-back original finite sequence to align these chart sequences, inserting only empty-center isomorphisms. The sheaf ideals, boundary labels and invariant values agree on overlaps because they are pullbacks of the same original data. Their ranges are subsets of the original finite ranges. At a nonempty inverse center the original maximum is attained in the image and is exactly the maximum on the pullback; centers missed by the entire image give only inserted identities. Hence, after deleting globally empty-center steps, the glued sequence is canonical in the extended sense of the definition, with at most $m$ blowups, even when $X^{\mathrm{\prime }}$ is not quasi-compact. Compatibility with further étale pullbacks follows by composition. Together with the descent and termination of step 5.1 and the semicontinuity of step 7.1, this proves all four stated conditions.