Smoothness survives base change and composition

We prove clause 1. Let $h:S^{\mathrm{\prime }}\to S$ be arbitrary and let $y\in X{\times }_S{S}^{\mathrm{\prime }}$ with images $x\in X$ and $s^{\mathrm{\prime }}\in S^{\mathrm{\prime }}$, and put $s=f(x)=h(s^{\mathrm{\prime }})$. Choose affine opens $V=\mathrm{Spec}A\subseteq S$ containing $s$, then $U=\mathrm{Spec}B\subseteq X$ containing $x$ with $f(U)\subseteq V$, and then $W=\mathrm{Spec}{A}^{\mathrm{\prime }}\subseteq S^{\mathrm{\prime }}$ containing $s^{\mathrm{\prime }}$ with $h(W)\subseteq V$.

We prove clauses 2 and 3. Let $y\in Y$ with images $x=g(y)\in X$ and $s=f(x)\in S$. Choose affine opens $V=\mathrm{Spec}A\subseteq S$ containing $s$, then $U=\mathrm{Spec}B\subseteq X$ containing $x$ with $f(U)\subseteq V$, and then $Z=\mathrm{Spec}C\subseteq Y$ containing $y$ with $g(Z)\subseteq U$.

By [F5] the fibre product $U{\times }_{V}W$ is the affine scheme $\mathrm{Spec}(B{\otimes }_A{A}^{\mathrm{\prime }})$, and by [F6] the canonical morphism $U{\times }_{V}W\to X{\times }_S{S}^{\mathrm{\prime }}$ is an open immersion: it factors as $U{\times }_{V}W\to X{\times }_{S}W\to X{\times }_S{S}^{\mathrm{\prime }}$, where the second arrow is a base change of the open immersion $W\subseteq S^{\mathrm{\prime }}$ and the first is a base change of the open immersion $U\subseteq X$ (using $U{\times }_{V}W=U{\times }_{S}W$ because $U\to S$ and $W\to S$ both factor through $V$), and a composite of open immersions is an open immersion. Hence $\mathrm{Spec}(B{\otimes }_A{A}^{\mathrm{\prime }})$ is identified with an affine open subscheme $U^{\mathrm{\prime }}\subseteq X{\times }_S{S}^{\mathrm{\prime }}$ containing $y$, and $f_{S^{\mathrm{\prime }}}(U^{\mathrm{\prime }})\subseteq W$.

By the forward direction of [F2] applied to $f$ at $x$ and to $g$ at $y$, the ring map $A\to B$ has a standard smooth presentation at the prime of $x$ and $B\to C$ has one at the prime of $y$. By the composition clause of [F3] the map $A\to C$ is standard smooth at the prime of $y$, and the composed presentation has relative dimension equal to the sum of the relative dimensions of the two presentations.

Since $f$ is smooth at $x$, [F2] provides, after shrinking the chart of step 1.1 if necessary, a standard smooth presentation of $B_h$ over $A$ at the prime $\mathfrak{q}\subset B$ of $x$, of some relative dimension $m-r$. By the base-change clause of [F3] the algebra $A^{\mathrm{\prime }}{\otimes }_{A}B=B{\otimes }_A{A}^{\mathrm{\prime }}$ is standard smooth over $A^{\mathrm{\prime }}$ at every prime lying over $\mathfrak{q}$, in particular at the prime ${\mathfrak{q}}^{\mathrm{\prime }}$ corresponding to $y\in U^{\mathrm{\prime }}$.

By the backward direction of [F2] applied with the chart $Z\to V$, the composite $f\circ g$ is smooth at $y$, and the chart of step 2.2 exhibits relative dimension equal to that sum at $y$.

The chart $U^{\mathrm{\prime }}=\mathrm{Spec}(B{\otimes }_A{A}^{\mathrm{\prime }})\to W=\mathrm{Spec}{A}^{\mathrm{\prime }}$ of steps 2.1 and 3.1 is an affine chart for $f_{S^{\mathrm{\prime }}}$ at $y$ whose ring map is standard smooth at ${\mathfrak{q}}^{\mathrm{\prime }}$, so the backward direction of the criterion [F2] gives that $f_{S^{\mathrm{\prime }}}$ is smooth at $y$. As $y$ was arbitrary, $f_{S^{\mathrm{\prime }}}$ is smooth, proving clause 1.

By [F4] each standard smooth chart at a smooth point exhibits the relative dimension of the corresponding smooth morphism at that point. Hence, writing $m={\mathrm{reldim}}_f(g(y))$ and $n={\mathrm{reldim}}_g(y)$, the two presentations chosen in step 2.2 have relative dimensions $m$ and $n$, their composite has relative dimension $m+n$, and this is ${\mathrm{reldim}}_{f\circ g}(y)$; this proves clause 3.

Finally, if $f$ and $g$ are smooth, then every $y\in Y$ satisfies the hypotheses of steps 1.2, 2.1, 2.2, 3.2 and 4.2, so $f\circ g$ is smooth at every point of $Y$, proving clause 2. The empty-source cases are vacuous: if $Y$, $X$ or $S^{\mathrm{\prime }}$ is empty the relevant assertions have no points to check. The Axiom of Choice [F8] is assumed in the Statement and used exactly through the criterion [F2], invoked in steps 3.1, 4.1, 2.2 and 3.2. [F1, F2, F8, step 4.2] $\mathrm{\square }$