Étale stability

Preservation of relative dimension under base change. Suppose $f$ is smooth at $x$; let $s=f(x)$ and let $y\in X{\times }_S{S}^{\mathrm{\prime }}$ lie over $x$, with image $s^{\mathrm{\prime }}$ in $S^{\mathrm{\prime }}$. By [F2] the base change $f_{S^{\mathrm{\prime }}}$ is smooth at $y$, so its relative dimension at $y$ is defined. We compare geometric fibres. By [F4] the fibre of $f_{S^{\mathrm{\prime }}}$ over $s^{\mathrm{\prime }}$ is canonically $X_s{\times }_{\mathrm{Spec}\kappa (s)}\mathrm{Spec}\kappa (s^{\mathrm{\prime }})$, and base changing further along any field extension $K\mathrm{/}\kappa (s^{\mathrm{\prime }})$ gives a canonical isomorphism $(X_{S^{\mathrm{\prime }}}{)}_{s^{\mathrm{\prime }},K}\cong X_s{\times }_{\mathrm{Spec}\kappa (s)}\mathrm{Spec}K,$ the right side being the base-changed fibre of $f$ along the composite field extension $K\mathrm{/}\kappa (s)$. Every point over the image of $y$ maps under this isomorphism to a point over the image of $x$, although not every point over $x$ need lie over this chosen $y$. At each of these points [F3] gives local dimension ${\mathrm{reldim}}_f(x)$. Thus every local dimension tested for $f_{S^{\mathrm{\prime }}}$ at $y$ has this value, and ${\mathrm{reldim}}_{f_{S^{\mathrm{\prime }}}}(y)={\mathrm{reldim}}_f(x)$.

Clause 1: base change. Assume $f$ is étale and let $y\in X{\times }_S{S}^{\mathrm{\prime }}$ over $x\in X$. Then $f$ is smooth at $x$ and ${\mathrm{reldim}}_f(x)=0$ by [F1]. By [F2] the base change $f_{S^{\mathrm{\prime }}}$ is smooth at $y$, and step 1.1 gives ${\mathrm{reldim}}_{f_{S^{\mathrm{\prime }}}}(y)=0$. By [F1] the base change is étale at $y$; since $y$ was arbitrary, $f_{S^{\mathrm{\prime }}}$ is étale. If $X$ is empty then so is $X{\times }_S{S}^{\mathrm{\prime }}$ and the conclusion is vacuous.

Clause 2: composition. Assume $f$ and $g$ are étale and let $y\in Y$, with $x=g(y)$. Then $f$ is smooth at $x$ and $g$ is smooth at $y$ by [F1], and their relative dimensions vanish. By [F2] the composite $f\circ g$ is smooth at $y$ and ${\mathrm{reldim}}_{f\circ g}(y)={\mathrm{reldim}}_f(x)+{\mathrm{reldim}}_g(y)=0+0=0.$ By [F1] the composite is étale at $y$; since $y$ was arbitrary, $f\circ g$ is étale.

Pointwise statements and accounting. The two pointwise assertions of clause 3 are exactly the arguments of steps 2.1 and 2.2 performed at one chosen point, and the global clauses follow by the arbitrary choice of the point. The Axiom of Choice [F6] is assumed in the Statement and is used exactly through the smooth stability theorem [F2], which is invoked in steps 1.1, 2.1 and 2.2; the fibre identification of step 1.1 uses only the canonical isomorphism [F4]. Empty sources are covered in step 2.1 and in the vacuous form of step 2.2. [F1, F4, F6, step 2.1, step 2.2] $\mathrm{\square }$