An automorphism of the completed local ring matching two tangent directions preserves the homogenization

Given: Assume $\mathrm{char}K=0$. Let $(\mathcal{I},E,\mu )$ be a marked ideal of maximal order on the smooth $K$-scheme $X$, let $x\in \mathrm{supp}(\mathcal{I},E,\mu )$, let $u,v\in T(\mathcal{I},\mu {)}_x={\mathcal{D}}^{\mu -1}(\mathcal{I}{)}_x$ be tangent directions transversal to $E$, and let $R={\widehat{\mathcal{O}}}_{X,x}$ with ${\widehat{X}}_x=\mathrm{Spec}R$.

Marked ideals of maximal order, tangent directions and transversality to the exceptional divisors, embedding dimension and regular local ring: $T(\mathcal{I})={\mathcal{D}}^{\mu -1}(\mathcal{I})$; $u,v\in T(\mathcal{I}{)}_x$ have multiplicity one; since $X$ is smooth at $x$ the local ring is regular and $u,v$ are each part of a regular system of parameters; transversality to $E$ means the parameters can be chosen compatible with the local equations of the members of $E$ through $x$.

Completion of a Noetherian local ring is local with the same residue field, completion preserves regular local rings: $R$ is a Noetherian regular local ring, faithful flat over ${\mathcal{O}}_{X,x}$, with maximal ideal $\mathfrak{m}R$; $\widehat{\mathcal{I}}=\mathcal{I}R$, and the completed tangent ideal is $\widehat{T}=T(\mathcal{I})R$.

The homogenized ideal of a marked ideal of maximal order: $H(\mathcal{I})={\sum }_{i=0}^{\mu -1}{\mathcal{D}}^i(\mathcal{I})T(\mathcal{I}{)}^i$, so its completion is $\widehat{H(\mathcal{I})}={\sum }_i\widehat{{\mathcal{D}}^i(\mathcal{I})}{\widehat{T}}^i$.

Field, Derivation of an algebra, Derivative ideals of an ideal sheaf and of a marked ideal: Write $\widehat{{\mathcal{D}}^i(\mathcal{I})}={\mathcal{D}}^i(\mathcal{I})R$; formal parameter derivatives fixing the coefficient field send this completed ideal into ${\mathcal{D}}^{i+1}(\mathcal{I})R$. To justify this, restrict a formal parameter derivation to ${\mathcal{O}}_{X,x}$. It is a $K$-derivation into $R$, and the finite-free differential module identifies it with an $R$-linear combination of the algebraic $K$-derivations (Differentials of a smooth morphism, Derivations are maps out of Ω). Leibniz also differentiates the multiplying coefficients in $R$, giving terms already in the lower derivative ideal. This supplies the Taylor containment without identifying the full algebraic differential module of a power-series ring with a finite module. In characteristic zero, for a continuous coordinate substitution $u_j\mapsto u_j+{\delta }_j$ with all ${\delta }_j$ in an ideal $T$, formal Taylor expansion gives ${\phi }^{\ast }(f)={\sum }_{\alpha \in {\mathbb{N}}^n}\frac{1}{\alpha !}({\mathrm{\partial }}^{\alpha }f){\delta }^{\alpha },$ convergently in the maximal-adic topology; if $f\in {\mathcal{D}}^i(\mathcal{I})$, then ${\mathrm{\partial }}^{\alpha }f\in {\mathcal{D}}^{i+\mathrm{\mid }\alpha \mathrm{\mid }}(\mathcal{I})$.

Simple normal crossings divisors and simultaneous normal crossings position, Marked ideals of maximal order, tangent directions and transversality to the exceptional divisors: the local equations of the components of $E$ through $x$ span a subspace $W\subseteq \mathfrak{m}\mathrm{/}{\mathfrak{m}}^2$. Transversality says that each of $u$ and $v$ is independent of $W$, so the boundary equations together with either direction can be completed separately to a regular system of parameters; no common complementary parameters are asserted.

Transcendental residue elements adjoin across a maximal subfield, Separable residue elements adjoin across a maximal subfield, A complete equicharacteristic Noetherian local ring is a power-series quotient, regular local rings are domains and cohen macaulay: $R$ is a complete equicharacteristic regular local ring, and smoothness in characteristic zero makes $\kappa (x)\mathrm{/}K$ a finitely generated separably generated extension. Starting from the image of $K$ in $R$, lift a separating transcendence basis and then the finite separable algebraic generators by these field-adjunction lemmas; this gives a coefficient field $k\subset R$ containing $K$. The continuous parameter map $k\mathrm{⟦}{U}_1,\dots ,U_n\mathrm{⟧}\to R$ is surjective by the Cohen presentation. Its kernel is zero: the regular-local associated-graded theorem identifies the graded map on the parameter classes with an isomorphism, and a nonzero series in the kernel would have a least nonzero homogeneous term mapping to zero, a contradiction. Thus any regular parameter system identifies $R$ with a formal power-series ring, where substitution by another parameter system with invertible cotangent matrix is a continuous $K$-algebra automorphism.