Étale morphisms are locally standard étale

Choose affine charts $V=\mathrm{Spec}A\subseteq S$ and $U_0=\mathrm{Spec}{B}_0\subseteq X$ around $f(x)$ and $x$, with $A\to B_0$ finitely presented. Let $\mathfrak{q}\subseteq B_0$ lie over $\mathfrak{p}\subseteq A$. By [F2] the map is quasi-finite at $\mathfrak{q}$; shrink to a principal open around $\mathfrak{q}$ on which it is quasi-finite everywhere. Applying [F3] to that finite-type algebra gives a finite $A$-algebra $T$ and an open immersion from the shrunk chart into $\mathrm{Spec}T$, identifying local rings at the selected point. The map $A\to T$ is therefore étale at the corresponding prime $\mathfrak{r}\subseteq T$, although it need not be étale elsewhere.

The fibre $T{\otimes }_A\kappa (\mathfrak{p})$ is finite dimensional over $\kappa (\mathfrak{p})$, so [F4] splits it into local Artinian factors. The factor at $\mathfrak{r}$ is the finite separable field $\kappa (\mathfrak{r})$ by [F2], because étaleness makes the selected zero-dimensional local fibre reduced. Choose a nonzero primitive element $\bar{t}$ for this field extension (take $\bar{t}=1$ when $\kappa (\mathfrak{r})=\kappa (\mathfrak{p})$) and take $\bar{t}=0$ in every other factor; lift this element of the fibre to $t\in T$. Put $C=A[t]\subseteq T$. The prime ${\mathfrak{q}}^{\mathrm{\prime }}=\mathfrak{r}\cap C$ has no other prime of $T$ above it in the closed fibre: the selected factor has a nonzero primitive value, whereas every other factor has value zero. The localization $T_{{\mathfrak{q}}^{\mathrm{\prime }}}$ is therefore local with maximal ideal $\mathfrak{r}{T}_{{\mathfrak{q}}^{\mathrm{\prime }}}$; moreover its closed fibre over $\mathfrak{p}$ is the selected field factor $\kappa (\mathfrak{r})$, generated by the image of $t$.

Since $T$ is finite over $A$, it is finite over $C$ as a module, and the map $C_{{\mathfrak{q}}^{\mathrm{\prime }}}\to T_{{\mathfrak{q}}^{\mathrm{\prime }}}$ is an injective local map of finite modules with the same residue field. The closed fibre $T_{{\mathfrak{q}}^{\mathrm{\prime }}}\mathrm{/}\mathfrak{p}{T}_{{\mathfrak{q}}^{\mathrm{\prime }}}=\kappa (\mathfrak{r})$ from step 2.1 is generated by $t$, hence also equals $T_{{\mathfrak{q}}^{\mathrm{\prime }}}\mathrm{/}{\mathfrak{q}}^{\mathrm{\prime }}{T}_{{\mathfrak{q}}^{\mathrm{\prime }}}$. Nakayama applied to the finite cokernel over the local ring $C_{{\mathfrak{q}}^{\mathrm{\prime }}}$ gives surjectivity, and the map is already injective. Thus the local rings are isomorphic. The finite cokernel is zero after inverting one element of $C\setminus {\mathfrak{q}}^{\mathrm{\prime }}$, so after a further principal shrinking the finite algebra $T$ and the monogenic subalgebra $C=A[t]$ coincide. Shrinking inside the open image of $U_0$ then identifies an affine neighbourhood of $x$ with a localization of $C$.

Construct a monic relation with invertible derivative. After step 3.1, work at the corresponding prime $\mathfrak{q}$ of $C=A[t]$; this algebra is étale there because it agrees with the original étale chart on a neighbourhood. Let $I=\ker (A[T]\to C)$ and let $Q\subseteq A[T]$ be the inverse image of $\mathfrak{q}$. The polynomial derivation computation in [F7] gives ${\mathrm{\Omega }}_{C\mathrm{/}A,\mathfrak{q}}\cong C_{\mathfrak{q}}\mathrm{/}(H^{\mathrm{\prime }}(t):H\in I)$; étaleness makes this module zero by [F1]. Thus finitely many $H_i\in I$ and coefficients $u_i\in C_{\mathfrak{q}}$ have ${\sum }_i{u}_i{H}_i^{\mathrm{\prime }}(t)=1$. Lift the $u_i$ to fractions in $A[T{]}_Q$ and clear a common denominator outside $Q$: a single $P_1\in I_Q$ has derivative a unit at $\mathfrak{q}$. Since $t$ is integral over $A$, there is a monic $P_0\in I$. Multiplying $P_1$ by a denominator outside $Q$ if necessary gives a global polynomial in $I$ whose derivative remains a unit at $\mathfrak{q}$; call it again $P_1$. For $N\ge 2$ with $N\mathrm{deg}{P}_0>\mathrm{deg}{P}_1$, the polynomial $P=P_0^N+P_1$ is monic, belongs to $I$, and satisfies $P^{\mathrm{\prime }}(t)=P_1^{\mathrm{\prime }}(t)$ because $P_0(t)=0$.

Put $C^{\mathrm{\prime }}=A[T]\mathrm{/}(P)$, with its surjection onto $C$, and localize at the selected prime $Q^{\mathrm{\prime }}=Q\mathrm{/}(P)$, over $\mathfrak{p}\subseteq A$. Let $h$ be the irreducible factor of $\bar{P}\in \kappa (\mathfrak{p})[T]$ selected by $Q^{\mathrm{\prime }}$. Because $P^{\mathrm{\prime }}(t)$ is a unit modulo $Q^{\mathrm{\prime }}$, $h$ occurs with multiplicity one in $\bar{P}$ and $h^{\mathrm{\prime }}\mathrm{\ne }0$ modulo $h$. Localizing the finite fibre ring at $(h)$ therefore removes all other factors and gives $C_{Q^{\mathrm{\prime }}}^{\mathrm{\prime }}\mathrm{/}\mathfrak{p}{C}_{Q^{\mathrm{\prime }}}^{\mathrm{\prime }}\cong \kappa (\mathfrak{p})[T]\mathrm{/}(h)=\kappa (\mathfrak{q})$, with no nilpotent thickening. Since $C$ is étale at $\mathfrak{q}$, its closed-fibre local ring $C_{\mathfrak{q}}\mathrm{/}\mathfrak{p}{C}_{\mathfrak{q}}$ is the same field. Thus the surjection between these fibre rings is an isomorphism. Let $K$ be the localized kernel $\ker (C_{Q^{\mathrm{\prime }}}^{\mathrm{\prime }}\to C_{\mathfrak{q}})$. The source $C_{Q^{\mathrm{\prime }}}^{\mathrm{\prime }}$ is flat over $A_{\mathfrak{p}}$ because $P$ is monic (so $C^{\mathrm{\prime }}$ is finite free over $A$) and localization preserves flatness; the target is flat over $A_{\mathfrak{p}}$ by étaleness. Tensoring the exact sequence $0\to K\to C_{Q^{\mathrm{\prime }}}^{\mathrm{\prime }}\to C_{\mathfrak{q}}\to 0$ with $\kappa (\mathfrak{p})$ remains exact, whence $K\mathrm{/}\mathfrak{p}K=0$. The kernel $K$ is finitely generated over $C_{Q^{\mathrm{\prime }}}^{\mathrm{\prime }}$: local finite presentation of the étale algebra $C$ says that, after localizing the polynomial map $A[T]\to C$ around $Q$, its relation ideal $I_Q$ is finitely generated; the kernel of $C_{Q^{\mathrm{\prime }}}^{\mathrm{\prime }}\to C_{\mathfrak{q}}$ is the quotient $I_Q\mathrm{/}(P)$, hence finite. This finite relation set also permits the later principal shrinking. Since $\mathfrak{p}{C}_{Q^{\mathrm{\prime }}}^{\mathrm{\prime }}$ lies in its maximal ideal, Nakayama [F4] yields $K=0$. Finite generation lets us invert one element outside $Q^{\mathrm{\prime }}$ so that $C^{\mathrm{\prime }}\to C$ is an isomorphism on a principal neighbourhood; invert also $P^{\mathrm{\prime }}(t)$. This gives the required standard étale chart by [F5], proving the forward direction. Conversely every standard étale chart is étale by [F5], and étaleness is local on source and target.

If $X=\mathrm{\varnothing }$ there is no point $x$ and the assertion is vacuous. The zero-degree polynomial case yields the empty chart, so no nonempty point lies there. AC is used through the quasi-finite factorization, primitive element and finite-prime results [F2]–[F4]; all other selections are finite. Both directions are established by steps 4.1 and 5.1.