Local algebra tools for elementary etale changes of classical varieties

By [F1], each displayed zero-dimensional chart is flat and every geometric fibre is regular of dimension zero. A finite-type zero-dimensional algebra over a field is Artinian, and a regular zero-dimensional local ring is a field. Thus the fibres are finite products of finite separable field extensions; at a classical point over $k$ the selected factor is $k$. Base change and composition preserve the presentation and relative dimension zero by [F1]. Open restrictions localize these presentations. At a selected local ring the flat map is local, hence faithfully flat by [F4]. If the base algebra $A$ is reduced, its injection into the finite product of the fraction fields of its minimal-prime quotients remains injective after tensoring with the flat chart. The resulting factors are reduced by the geometric-fibre assertion. The chart is therefore reduced, as a subring of a product of reduced rings.

We justify openness on prime spectra before restricting to classical points. For an injective map of finite-type $k$-domains $R\hookrightarrow S$, generic affine normalization [F3] gives $0\mathrm{\ne }a\in R$ such that $S_a$ is finite and integral over an injected polynomial ring $R_a[z_1,\dots ,z_r]$. Each prime $\mathfrak{p}$ of $R_a$ extends to the prime $\mathfrak{p}{R}_a[z]$ and lifts to $S_a$ by lying over. Thus the spectral image contains $D(a)$ in its image closure. For a general affine source, pass to its finitely many minimal-prime quotients and the corresponding image closures. On each irreducible component remove the inverse image of such a principal image open and induct on the proper closed source complement. Noetherian induction terminates this argument and proves constructibility of the whole image. An open source has a finite principal-open cover, so the same argument applies to images of opens. Hence the image of any open under a finite-type map of the present Noetherian affine spaces is constructible. By going down, the image under a flat map is stable under generization: apply [F3] to the localization at a source point in that open. A constructible generization-stable subset of a Noetherian spectrum is open. Indeed its complement is constructible and specialization-stable; write that complement as a finite union of locally closed subsets, include the generic points of their irreducible closures, and specialization stability then includes their entire closures. The complement is closed. Thus the chart map is open. This remains true after base change between the present finite-type algebras. Restricting to closed $k$-points gives openness of the classical map: a nonempty fibre over a closed point has a closed point because it is a finite-type $k$-algebra.

We prove the monic local form at a classical point $q$ of a chart $A\to S$. By step 1.1 this is quasi-finite everywhere. Apply [F2] at $q$ to obtain $g$ in the relative integral closure $C\subseteq S$, nonzero at $q$, with $C_g=S_g$. Choose finite $A$-algebra generators of $S_g$, express them with numerators in $C$ and powers of $g$ as denominators, and let $T\subseteq C$ be generated by those numerators and $g$. Then $T$ is finite over $A$ and $T_g=S_g$. Thus an affine neighbourhood of $q$ is open in $\mathrm{Spec}T$. The fibre $T{\otimes }_{A}k$ is Artinian. Its selected local factor is $k$, because its localization agrees with the selected fibre local ring of $S$. Choose $t\in T$ whose image is $1$ in that factor and $0$ in every other factor; lifting is possible because the residue field of the base point is $k$. Put $D=A[t]\subseteq T$. The selected prime $q_D$ has exactly one prime of $T$ above it, because the selected nonzero value separates it from all other fibre factors. Consequently $T_{q_D}$ is local, finite over $D_{q_D}$, and its closed fibre is generated by $t$ and equals $k$. Nakayama applied to the finite cokernel of $D_{q_D}\hookrightarrow T_{q_D}$ makes that injection an isomorphism. Vanishing of its finite cokernel after one further localization identifies the original chart near $q$ with a localization of the monogenic finite algebra $D$.

Let $I=\ker (A[T]\to D)$ and let $Q$ be the selected prime of $A[T]$. The localized relative differentials of $D$ vanish by [F5] and the preceding identification. Thus a finite linear combination of derivatives of elements of $I$ is a unit at $Q$. Lift the coefficients to fractions of polynomials, clear a denominator outside $Q$, and sum the coefficient multiples of these relations. The derivatives of the coefficients multiply relations vanishing in $D$, so this produces $P_1\in I$ with $P_1^{\mathrm{\prime }}(t)$ a unit at the selected point. Since $t$ is integral, choose monic $P_0\in I$. For $N\ge 2$ and $N\mathrm{deg}{P}_0>\mathrm{deg}{P}_1$, put $P=P_0^N+P_1$. It is monic, belongs to $I$, and $P^{\mathrm{\prime }}(t)=P_1^{\mathrm{\prime }}(t)$ is a unit. The selected local fibre of $D^{\mathrm{\prime }}=A[T]\mathrm{/}(P)$ is a field: the selected root is simple, so localizing the finite polynomial fibre removes every other factor and leaves $k$. The fibre surjection $D_{Q\mathrm{/}(P)}^{\mathrm{\prime }}\to D_{q_D}$ is consequently an isomorphism.

Both local rings in step 3.1 are flat over the local base: the source is a localization of the finite free monic algebra $D^{\mathrm{\prime }}$, and the target agrees with the original flat chart. Tensor the kernel sequence with the base residue field; flatness of the target gives an injection on the kernel, and the identical fibres give $K\mathrm{/}\mathfrak{m}K=0$. The kernel is finite because the rings are Noetherian. Nakayama yields $K=0$. Finite generation permits a principal shrinking on which $D^{\mathrm{\prime }}\to D$ is an isomorphism; shrink inside the original chart and invert also $P^{\mathrm{\prime }}$. This is the desired monic presentation. Such neighbourhoods at all closed points cover the chart, since a nonempty closed subset of a finite-type $k$-space has a closed point. Conversely a monic presentation with invertible derivative is one of the square-Jacobian presentations of [F1]. Steps 1.1 and 2.1 give all the remaining assertions.