Higher-Dimensional Resolution of Singularities

Let $X$ be a locally Noetherian scheme, $\mathcal{I}\subseteq {\mathcal{O}}_X$ a coherent ideal sheaf (Coherent module sheaves, Ideal sheaves) and $x\in X$ with maximal ideal ${\mathfrak{m}}_x\subset {\mathcal{O}}_{X,x}$ (A local ring is a nonzero commutative ring with a unique maximal ideal). The order of $\mathcal{I}$ at $x$ is ${\mathrm{ord}}_x(\mathcal{I}):=\max \{n\ge 0:{\mathcal{I}}_x\subseteq {\mathfrak{m}}_x^n\},$ with ${\mathrm{ord}}_x(\mathcal{I}):=+\mathrm{\infty }$ when ${\mathcal{I}}_x=0$; the maximum is attained for ${\mathcal{I}}_x\mathrm{\ne }0$ because ${\bigcap }_n{\mathfrak{m}}_x^n=0$ in the Noetherian local ring ${\mathcal{O}}_{X,x}$ (The Krull intersection is the $(1-a)$-torsion submodule, and it vanishes in the Jacobson-radical case). For $f\in {\mathcal{O}}_X(U)$ put ${\mathrm{ord}}_x(f):={\mathrm{ord}}_x(f{\mathcal{O}}_U)$; a germ is of multiplicity one at $x$ if ${\mathrm{ord}}_x(f)=1$. If in addition ${\mathcal{O}}_{X,x}$ is regular — the only case used on this page — then $f$ is part of a regular system of parameters of ${\mathcal{O}}_{X,x}$, so the zero scheme $V(f)$ is regular at $x$ of dimension $\dim {\mathcal{O}}_{X,x}-1$ (regular local quotient by parameter is regular, under AC The Axiom of Choice). The parameter assertion follows by extending the nonzero class of $f$ in the finite-dimensional cotangent space to a basis and applying the finite-generator Nakayama argument. If $x\in V(\mathcal{I})$ and $s=\dim {\mathcal{O}}_{X,x}$, then ${\mathrm{ord}}_x(\mathcal{I})\ge 1$; ${\mathrm{ord}}_x$ is the multiplicity used throughout this page, and it differs from the Hilbert-Samuel multiplicity, which is not used here.

Let $X$ be a regular locally Noetherian scheme of pure dimension $n$ (embedding dimension and regular local ring). A reduced effective Cartier divisor $D\subseteq X$ (Effective cartier divisor, Cartier divisor) is a strict normal crossings divisor (an SNC divisor) if for every point $p\in X$, writing $d_p=\dim {\mathcal{O}}_{X,p}$, there is a regular system of parameters $u_1,\dots ,u_{d_p}$ of ${\mathcal{O}}_{X,p}$ and a subset $J\subseteq \{1,\dots ,d_p\}$ such that the ideal of $D$ in ${\mathcal{O}}_{X,p}$ is generated by ${\prod }_{j\in J}{u}_j$; equivalently, every irreducible component of $D$ is regular and the local equations of the components through $p$ extend to a regular system of parameters. A finite family $E=\{D_1,\dots ,D_m\}$ of reduced effective Cartier divisors on $X$ is in simultaneous SNC position if for every subset $J\subseteq \{1,\dots ,m\}$ the union of components of the divisors in $J$ is an SNC divisor; equivalently, at every point $p$ the union of all components of members of $E$ passing through $p$ is cut out by a subset of a regular system of parameters of ${\mathcal{O}}_{X,p}$. For the marked-ideal algorithm on this page, the irreducible components within each individual member of $E$ must be pairwise disjoint, and no irreducible component is repeated in different members. Thus at a point each member contributes at most one distinct parameter equation. This is the additional boundary convention of Włodarczyk, Definition 2.1.1. Such a family $E$ always carries a fixed total order, and the members of $E$ play the role of the exceptional divisors of a multiple test blow-up.

Let $\phi \text{\hspace{0.17em}⁣}:X^{\mathrm{\prime }}\to X$ be a morphism of locally Noetherian schemes that is étale at $x^{\mathrm{\prime }}\in X^{\mathrm{\prime }}$ (Étale morphism of schemes), with $x=\phi (x^{\mathrm{\prime }})$, and let ${\mathfrak{m}}_x\subset {\mathcal{O}}_{X,x}$ and ${\mathfrak{m}}_{x^{\mathrm{\prime }}}\subset {\mathcal{O}}_{X^{\mathrm{\prime }},x^{\mathrm{\prime }}}$ be the maximal ideals (A local ring is a nonzero commutative ring with a unique maximal ideal). Then:

(1) $\phi$ is flat at $x^{\mathrm{\prime }}$ (Smooth morphism of schemes);

(2) ${\mathfrak{m}}_x{\mathcal{O}}_{X^{\mathrm{\prime }},x^{\mathrm{\prime }}}={\mathfrak{m}}_{x^{\mathrm{\prime }}}$;

(3) for every ideal $I\subseteq {\mathcal{O}}_{X,x}$ and every $n\ge 0$, $I\subseteq {\mathfrak{m}}_x^n⟺I{\mathcal{O}}_{X^{\mathrm{\prime }},x^{\mathrm{\prime }}}\subseteq {\mathfrak{m}}_{x^{\mathrm{\prime }}}^n\mathrm{.}$

If the induced residue-field map $\kappa (x)\to \kappa (x^{\mathrm{\prime }})$ is an isomorphism, then the induced map of maximal-adic completions ${\widehat{\mathcal{O}}}_{X,x}⟶{\widehat{\mathcal{O}}}_{X^{\mathrm{\prime }},x^{\mathrm{\prime }}}$ is an isomorphism. This completion conclusion requires the residue-field hypothesis.

• The residue-field condition in the completion clause cannot be omitted. For a nontrivial finite separable extension $L\mathrm{/}K$, the morphism $\mathrm{Spec}L\to \mathrm{Spec}K$ is étale (Finite field extensions and etaleness), while the completed local rings are $L$ and $K$, and the induced completion map is the proper inclusion $K\hookrightarrow L$, hence is not an isomorphism. No assertion about abstract isomorphism of the two fields is needed. Włodarczyk's phrase “formal analytic isomorphism” in the proof of Lemma 2.4.1 is valid in its algebraically closed closed-point setting; the order argument for arbitrary étale points needs only (1)–(3).
• Flatness and the associated-graded argument prove the filtration and equal-residue completion claims without a choice principle.

Assume the Axiom of Choice. Let $K$ be a field of characteristic zero and let ${\iota }_U\text{\hspace{0.17em}⁣}:U\hookrightarrow {\mathbb{A}}_K^m$ and ${\iota }_V\text{\hspace{0.17em}⁣}:V\hookrightarrow {\mathbb{A}}_K^n$ be closed immersions of affine $K$-varieties of finite type (Closed immersions of schemes, Integral schemes). Let $\phi \text{\hspace{0.17em}⁣}:U\to V$ be étale at a $K$-rational point $u_0$, and put $v_0=\phi (u_0)$. Choose affine coordinates on the two ambient spaces carrying $u_0$ and $v_0$ to their origins.

Then there is a closed subscheme $X\subseteq {\mathbb{A}}_K^n{\times }_K{\mathbb{A}}_K^m$ containing the graph copy of $U$, smooth at $(v_0,u_0)$, such that the projection $\mathrm{\Phi }:={\mathrm{pr}}_1{\mathrm{\mid }}_X\text{\hspace{0.17em}⁣}:X\to {\mathbb{A}}_K^n$ is étale at $(v_0,u_0)$ and restricts on $U$ to ${\iota }_V\circ \phi$. Thus $\phi$ extends to a morphism of smooth ambient varieties étale at the chosen point.

After extending the ground field so the point is rational, the usual local factorization of a smooth morphism as an étale morphism followed by a projection gives the corresponding smooth ambient extension locally. This is the reduction used in the source's proof of functoriality for smooth morphisms.

• The source proof uses coordinates on ${\mathbb{A}}^n\times {\mathbb{A}}^m$, hence the ambient dimension is $n+m$. Its printed line saying $X\subset {\mathbb{A}}^m$ is inconsistent with those coordinates and with its projection ${\mathbb{A}}^{m+n}\to {\mathbb{A}}^n$; the statement above follows the coordinate proof.
• The ambient extension is proved for the source's affine-embedding and rational-point setting. The smooth-morphism consequence is used after passage to a geometric point; no claim is made here that an arbitrary non-rational point over the original field is rational.

This page follows the source conventions of Field and A field's prime subfield is isomorphic to $\mathbb{Q}$ in characteristic zero and to ${\mathbb{F}}_p$ in characteristic $p$: $K$ is a field of characteristic zero, i.e. its prime subfield is $\mathbb{Q}$. A $K$-variety is an integral, separated scheme of finite type over $K$ (Integral schemes, Separated morphism of schemes, Locally finite type and finite type morphisms); a smooth $K$-scheme is a scheme of finite type over $K$ whose structure morphism is smooth (Smooth morphism of schemes); in characteristic zero a scheme of finite type over $K$ is smooth over $K$ exactly when all its local rings are regular, by the perfect-field geometric-regularity criterion (Field tests for geometric regularity, Locally standard smooth iff flat with geometrically regular fibres). The core resolution construction is over an algebraically closed field of characteristic zero, as in the source. Items with an explicitly broader field or characteristic range retain their stated hypotheses; the descent item and final theorems apply over arbitrary characteristic-zero fields. The positive-characteristic derivative converses require the perfect-field qualifications stated in their items. All blowups are blowups of regular closed subschemes of smooth $K$-schemes, and all divisors are effective Cartier divisors; SNC' abbreviates simple normal crossings' in the sense of Simple normal crossings divisors and simultaneous normal crossings position. Resolutions are stated for reduced or integral base schemes and are constructed from an ambient smooth scheme. The Axiom of Choice is assumed throughout this resolution development and is inherited from the published blowup and relative-Proj suppliers (The Axiom of Choice); no dependent-choice or other choice principle is used by this page's new arguments beyond what those suppliers already assume.

Assume AC (The Axiom of Choice), as inherited from the regular-local algebra suppliers.

Let $\phi \text{\hspace{0.17em}⁣}:X^{\mathrm{\prime }}\to X$ be a smooth morphism of smooth $K$-schemes (Smooth morphism of schemes).

(1) For every coherent ideal sheaf $\mathcal{I}\subseteq {\mathcal{O}}_X$ (Coherent module sheaves) and every $x^{\mathrm{\prime }}\in X^{\mathrm{\prime }}$ with $x=\phi (x^{\mathrm{\prime }})$, ${\mathrm{ord}}_{x^{\mathrm{\prime }}}({\phi }^{\ast }\mathcal{I})={\mathrm{ord}}_x(\mathcal{I}),$ where the order is that of Order of an ideal sheaf at a point.

(2) Assume in addition that $X^{\mathrm{\prime }}$ is of pure dimension, as required by the SNC interface. If $E$ is a family of divisors in simultaneous SNC position on $X$ (Simple normal crossings divisors and simultaneous normal crossings position) then the family ${\phi }^{-1}(E)$ of scheme-theoretic inverse images of its members is a family of divisors in simultaneous SNC position on $X^{\mathrm{\prime }}$; the individual inverse images are reduced effective Cartier divisors (Effective cartier divisor).

This is the source's Lemma 2.4.1.

Let $X$ be a smooth $K$-scheme of pure dimension $n$ (Conventions for the resolution development) and let $E$ be a finite family of reduced divisors on $X$ in simultaneous SNC position (Simple normal crossings divisors and simultaneous normal crossings position). A marked ideal on $X$ is a triple $(\mathcal{I},E,\mu )$ consisting of a coherent ideal sheaf $\mathcal{I}\subseteq {\mathcal{O}}_X$, the family $E$, and an integer $\mu \ge 0$; it is displayed as $(\mathcal{I},\mu )$ when $E$ is understood. The support of $(\mathcal{I},E,\mu )$ is $\mathrm{supp}(\mathcal{I},E,\mu ):=\{x\in X:{\mathrm{ord}}_x(\mathcal{I})\ge \mu \}$ (Order of an ideal sheaf at a point); it is in fact a closed subset of $X$, as shown below. The canonical-resolution existence theorem below requires $\mu \ge 1$ and $\mathcal{I}$ nonzero at the generic point of every irreducible component of $X$. Merely requiring $\mathcal{I}\mathrm{\ne }0$ globally does not suffice on a disconnected smooth scheme. For $\mu =0$ the support is all of $X$, so support-clearing resolution by the canonical algorithm is not asserted. Writing $(\mathcal{I},\mu )$ for $(\mathcal{I},E,\mu )$ suppresses only $E$, never the order $\mu$.

Let $(\mathcal{I},E,\mu )$ be a marked ideal on the smooth $K$-scheme $X$ (Marked ideals and their support). A multiple test blow-up of $(\mathcal{I},E,\mu )$ is a finite sequence $X=X_0\stackrel{{\sigma }_1}{\leftarrow }{X}_1\leftarrow \cdots \leftarrow X_r$ in which each ${\sigma }_i\text{\hspace{0.17em}⁣}:X_i\to X_{i-1}$ is specified either as an inserted isomorphism step or as the blowup (Blowup of a scheme along an ideal sheaf) of $X_{i-1}$ at a regular closed subscheme $C_{i-1}\subseteq \mathrm{supp}({\mathcal{I}}_{i-1},E_{i-1},\mu )$ (Closed immersions of schemes) that has SNC with $E_{i-1}$, together with the marked ideals defined inductively for a blowup step by $({\mathcal{I}}_i,E_i,\mu ),{\mathcal{I}}_i:=\mathcal{I}(D_i{)}^{-\mu }\cdot {\sigma }_i^{\ast }({\mathcal{I}}_{i-1}),E_i:={\sigma }_i^{\mathrm{c}}(E_{i-1})\cup \{D_i\},$ where $D_i\subseteq X_i$ is the exceptional divisor (Exceptional subscheme of a blowup), $\mathcal{I}(D_i)$ is its invertible ideal (Invertible sheaf of cartier divisor), $\mathcal{I}(D_i{)}^{-\mu }$ its $(-\mu )$-th tensor power (Invertible sheaves, Tensor product of sheaves of modules), and ${\sigma }_i^{\mathrm{c}}(E_{i-1})$ is the family of strict transforms (Strict transform of a closed subscheme) ordered so that all old members precede $D_i$. Empty members of the transformed boundary are omitted, with the order of all surviving labels retained. The transform ${\sigma }_i^{\mathrm{c}}({\mathcal{I}}_{i-1},\mu ):=({\mathcal{I}}_i,\mu )$ is the controlled transform of $({\mathcal{I}}_{i-1},\mu )$; for a local section $f\in {\mathcal{I}}_{i-1}(U)$ with local equation $y_i$ of $D_i$ the section $y_i^{-\mu }{\sigma }_i^{\ast }(f)$ is a controlled transform of $f$, well defined up to a unit. A blowup step retains the specified center and $D_i={\sigma }_i^{-1}(C_{i-1})$, even when its underlying morphism is an isomorphism, as for a Cartier center. An inserted isomorphism step has empty $D_i$, transports the ideal and ordered boundary, and appends no boundary member. A resolution of $(\mathcal{I},E,\mu )$ is a multiple test blow-up with $\mathrm{supp}({\mathcal{I}}_r,E_r,\mu )=\mathrm{\varnothing }$. An extension of a multiple test blow-up $(X_i{)}_{0\le i\le m}$ is a multiple test blow-up $(X_j^{\mathrm{\prime }}{)}_{0\le j\le m^{\mathrm{\prime }}}$ with $X_0^{\mathrm{\prime }}=X$, indices $j_0=0<j_1<\cdots <j_m$ and isomorphisms forming an identification $X_{j_i}^{\mathrm{\prime }}=X_i$; the extended sequence is obtained from the original one by inserting isomorphisms, with every original blow-up retained in its original order. No new nontrivial blow-ups are inserted. This is the extension convention of the source, Definition 2.1.5.

Assume AC (The Axiom of Choice) through the smooth-differential supplier. Let $X$ be a smooth $K$-scheme (Smooth morphism of schemes) with sheaf of relative differentials ${\mathrm{\Omega }}_{X\mathrm{/}K}$ (Sheaf of relative Kähler differentials, Existence and generators of Kähler differentials) and derivation sheaf ${\mathrm{Der}}_K({\mathcal{O}}_X)$ (Derivation of an algebra, Derivations are maps out of Ω). For a coherent ideal sheaf $\mathcal{I}\subseteq {\mathcal{O}}_X$ (Coherent module sheaves) define the first derivative (extension) $\mathcal{D}(\mathcal{I})$ intrinsically as the ideal generated by $\mathcal{I}$ and all $D(f)$, for local sections $f$ of $\mathcal{I}$ and $K$-derivations $D$. The smooth-differential theorem Differentials of a smooth morphism makes ${\mathrm{\Omega }}_{X\mathrm{/}K}$ finite locally free. Choose functions $u_1,\dots ,u_n$ whose differentials form a basis on a neighbourhood and let $\mathrm{\partial }\mathrm{/}\mathrm{\partial }{u}_i$ be the dual derivations. Here $n$ is the relative dimension over $K$, which need not equal the local-ring dimension at a nonclosed point. For generators $f_1,\dots ,f_s$ of $\mathcal{I}$, the same ideal is generated by the $f_j$ and $\mathrm{\partial }{f}_j\mathrm{/}\mathrm{\partial }{u}_i$: Leibniz reduces derivatives of arbitrary $\sum a_j{f}_j$ to these generators, and every derivation is a linear combination of the dual basis. This finite generator list proves coherence and independence of generators and differential coordinates. Set ${\mathcal{D}}^0(\mathcal{I}):=\mathcal{I}$ and ${\mathcal{D}}^i(\mathcal{I}):=\mathcal{D}({\mathcal{D}}^{i-1}(\mathcal{I}))$. The ideal is generated by $f_j$ and their derivatives of order at most $i$ and is independent of the chosen generators and coordinates; for a marked ideal put ${\mathcal{D}}^i(\mathcal{I},\mu ):=({\mathcal{D}}^i(\mathcal{I}),\mu -i)$ for $0\le i\le \mu$. In characteristic $p>0$, ordinary derivations need not lower order: on ${\mathbb{A}}_K^1$ for a field $K$ of characteristic $p$, the ideal $\mathcal{I}=(x^p)$ satisfies $\mathcal{D}(\mathcal{I})=\mathcal{I}$.

Two marked ideals $(\mathcal{I},E_{\mathcal{I}},{\mu }_{\mathcal{I}})$ and $(\mathcal{J},E_{\mathcal{J}},{\mu }_{\mathcal{J}})$ on the same smooth $K$-scheme $X$ are equivalent, $(\mathcal{I},E_{\mathcal{I}},{\mu }_{\mathcal{I}})\simeq (\mathcal{J},E_{\mathcal{J}},{\mu }_{\mathcal{J}}),$ if: (1) $E_{\mathcal{I}}=E_{\mathcal{J}}$ as ordered families of divisors; (2) their supports agree; and (3) the multiple test blow-ups of the one are exactly the multiple test blow-ups of the other, and for every such blow-up $(X_i)$ the induced supports agree, $\mathrm{supp}({\mathcal{I}}_i,E_i,{\mu }_{\mathcal{I}})=\mathrm{supp}({\mathcal{J}}_i,E_i,{\mu }_{\mathcal{J}})$ for every $i$ (Multiple test blow-ups, controlled transforms and resolutions of marked ideals). The relation is reflexive, symmetric and transitive by definition, and the algorithm below replaces a marked ideal by equivalent ones at the steps marked in the source.

Assume AC (The Axiom of Choice), inherited from the regular-local and blowup suppliers. Let $(\mathcal{I},E,\mu )$ be a marked ideal on a smooth $K$-scheme $X$, let $C\subseteq \mathrm{supp}(\mathcal{I},E,\mu )$ be a regular closed subscheme with SNC with $E$, let $\sigma \text{\hspace{0.17em}⁣}:X^{\mathrm{\prime }}\to X$ be the blowup of $C$ with exceptional divisor $D$ (Blowup of a scheme along an ideal sheaf, Exceptional subscheme of a blowup), and let ${\mathcal{I}}_C\subseteq {\mathcal{O}}_X$ be the ideal sheaf of $C$ and $\mathcal{I}(D)\subseteq {\mathcal{O}}_{X^{\mathrm{\prime }}}$ the invertible ideal of $D$ (Invertible sheaf of cartier divisor). Then $\mathcal{I}\subseteq {\mathcal{I}}_C^{\mu }\text{and}{\sigma }^{\ast }\mathcal{I}\subseteq \mathcal{I}(D{)}^{\mu }\mathrm{.}$ Consequently the controlled transform ${\sigma }^{\mathrm{c}}(\mathcal{I},\mu ):=(\mathcal{I}(D{)}^{-\mu }{\sigma }^{\ast }\mathcal{I},\mu )$ is an ideal sheaf on $X^{\mathrm{\prime }}$ (Multiple test blow-ups, controlled transforms and resolutions of marked ideals), and for $f\in \mathcal{I}(U)$ the local section $y^{-\mu }{\sigma }^{\ast }(f)$, $y$ a local equation of $D$, is well defined up to a unit and generates the controlled transform wherever $f$ generates $\mathcal{I}$.

• The two inclusions admit the empty center reading: if $C=\mathrm{\varnothing }$ then ${\mathcal{I}}_C={\mathcal{O}}_X$ and both inclusions are trivial; the controlled transform is then ${\sigma }^{\ast }\mathcal{I}$ with $\sigma$ an isomorphism, matching the definition of a multiple test blow-up extended by isomorphisms.
• No choice beyond the published blowup interface is used in this item.

Assume AC (The Axiom of Choice) for the completed-local and smooth-coordinate arguments.

Let $(\mathcal{I},\mu )$ be a marked ideal on a smooth $K$-scheme $X$ with $\mathcal{I}\mathrm{\ne }0$ (Marked ideals and their support, Smooth morphism of schemes), and let $0\le i\le \mu -1$ (Derivative ideals of an ideal sheaf and of a marked ideal). In every characteristic, $\mathrm{supp}(\mathcal{I},\mu )\subseteq \mathrm{supp}({\mathcal{D}}^i(\mathcal{I}),\mu -i),$ and, when $K$ is perfect, $\mathrm{supp}(\mathcal{I},\mu )$ is closed. If $K$ has characteristic zero, or is perfect of characteristic $p>0$ with $\mu <p$ (Field), then the inclusion is an equality. In these same characteristics, for $\mu \ge 1$ the condition ${\mathrm{ord}}_x(\mathcal{I})\le \mu$ for every $x\in X$ is equivalent to ${\mathcal{D}}^{\mu }(\mathcal{I})={\mathcal{O}}_X$.