Resolution of singularities in characteristic zero

Local construction. Choose a finite affine cover $Y={\bigcup }_i{U}_i$ with closed embeddings $U_i\hookrightarrow X_i$ into smooth affine $K$-schemes. By [F1] each strict transform ${\tilde{U}}_i$ is smooth and gives a proper birational morphism to $U_i$ which is the identity over its smooth locus. By [F2] this resolution is independent of its ambient embedding.

An intrinsic field of constants. Put $R=\mathrm{\Gamma }(Y,{\mathcal{O}}_Y)$, embedded in the function field using any nonempty affine chart. The elements of $R$ algebraic over $K$ form a field $L$: sums and products remain algebraic, and the minimal polynomial of a nonzero algebraic element expresses its inverse as a polynomial in that element. For a nonempty affine chart $\mathrm{Spec}A$, [F9] makes $A$ finite over $K[z_1,\dots ,z_d]$. If $L_0\mathrm{/}K$ is any finite subextension of $L$, the $z_i$ remain algebraically independent over $L_0$ and $L_0(z_1,\dots ,z_d)$ embeds in the finite-dimensional algebra $A{\otimes }_{K[z]}K(z)$. Thus $[L_0:K]$ is bounded by the number of module generators of $A$. Among these degrees choose a maximal one; adjoining any further element of $L$ cannot increase it, so $L\mathrm{/}K$ is finite, hence separable.

Gluing. On $U_i\cap U_j$, [F3], applied on affine open subcharts, identifies the two restrictions. These identifications satisfy the cocycle condition: each is the identity over the dense smooth locus, and two morphisms from an integral reduced scheme to a separated scheme agreeing on a dense open agree everywhere (on affine target charts the difference of each pair of pulled-back functions vanishes in the function field). Gluing gives $\tilde{Y}\to Y$. Smoothness and finite type are local on these charts; properness is local on the target and follows from [F5]. Birationality and the isomorphism over the smooth locus follow from step 1.1. The same unique comparisons show independence of the affine cover.

Étale comparison. First work over an algebraic closure of $K$. Geometric charts can be reduced with several irreducible components. The proofs of [F2] and [F6] still apply to them: the former uses only coordinate generators and polynomial ambient automorphisms, and the latter uses the completed local-ring isomorphism and graph equations, with no integrality assumption. Comparisons are unique on a reduced source by checking equality on the dense smooth open of every component. At each closed point of an étale morphism $V\to U$, choose affine charts and use [F6] to realize it as a Cartesian restriction of an étale ambient map $X^{\mathrm{\prime }}\to X$. Its defining ideal is the pullback of that of $U$. By [F7] the ambient centers and marked transforms pull back stage by stage. Blowups commute with flat base change: their Rees algebras pull back because flatness preserves every inclusion of an ideal power. Strict transforms commute as well: saturation by an exceptional equation is a filtered union of kernels of multiplication maps, all preserved by flat pullback. Thus the modified embedded procedure has the same strict transforms and stopping decisions after pullback, omitting only empty centers, and its final strict transform is $\tilde{U}{\times }_{U}V$. By [F2] this is the canonical resolution of $V$. Closed points cover the comparison loci by open neighbourhoods, since a finite-type scheme over an algebraically closed field has a closed point in every nonempty closed subset. By [F4] the canonical centers over $K$ are the descended geometric centers; equality of their ideals and the resulting comparison descend by faithful flatness. This proves the étale comparison over $K$.

Every field subring of $R$ lies in $L$. Otherwise it contains an element $f$ transcendental over $K$. Its prime field is $\mathbb{Q}$, so every $f-q$, $q\in \mathbb{Q}$, is a unit in $R$ and in $A$. The nonzero finite-type $K(f)$-algebra $A{\otimes }_{K[f]}K(f)$ has a maximal ideal with finite residue field $E\mathrm{/}K(f)$ by [F9]. Write $b_i\in E$ for the images of finite $K[f]$-algebra generators of $A$. Clearing denominators in their monic equations over $K(f)$ gives a nonzero polynomial $p$ such that $B=K[f,1\mathrm{/}p,b_1,\dots ,b_s]$ is integral over $K[f,1\mathrm{/}p]$, and there is a map $A\to B$. Choose $q\in \mathbb{Q}$ with $p(q)\mathrm{\ne }0$. Lying over supplies a prime of $B$ above $(f-q)$, contradicting the image of the unit $f-q\in A$. Hence every field subring is contained in $L$, so $L$ is the unique largest field subring of $R$ and is preserved by every scheme automorphism of $Y$.

Smooth comparison. For $U\times {\mathbb{A}}^r\to U$, pull back a chosen embedded presentation along $X\times {\mathbb{A}}^r\to X$. The same center, blowup and strict-transform calculations of step 2.2 give the canonical resolution $\tilde{U}\times {\mathbb{A}}^r$. A smooth germ factors as an étale map to $U\times {\mathbb{A}}^r$ followed by projection, so step 2.2 and this product case give ${\tilde{Y}}^{\mathrm{\prime }}\cong \tilde{Y}{\times }_Y{Y}^{\mathrm{\prime }}$ locally for every smooth $Y^{\mathrm{\prime }}\to Y$. The comparisons glue by the uniqueness argument of step 2.1, and their composite for two smooth maps is the same canonical comparison. The lifted map to $\tilde{Y}$ is smooth because it is the base change of $Y^{\mathrm{\prime }}\to Y$.

Equivariance. Regard $Y$ as an $L$-variety using $L\subseteq R$; it is still integral, separated and finite type. Its canonical resolution over $L$ equals the one over $K$: choose smooth affine $L$-ambient presentations, which are smooth over $K$ because $L\mathrm{/}K$ is finite separable, and use [F2]. In these ambients every $K$-derivation kills $L$ (differentiate the separable minimal polynomial), so the $K$- and $L$-derivative ideals coincide. Orders, boundary components, homogenized and coefficient ideals, monomial parts and companion ideals therefore coincide, as do the invariant centers and the modified embedded stopping rule. Each scheme automorphism of $Y$ is semilinear over its induced automorphism of $L$ by step 2.3; [F8] then transports each center and lifts it through the blowups and gluing. The lift is unique by step 2.1, so lifts preserve identity and composition and yield a group action on $\tilde{Y}$. This proves equivariance even when the original action does not preserve $K$, and step 3.1 proves the promised smooth functoriality.