Relative differentials commute with scheme base change

The canonical map. The composite $D\text{\hspace{0.17em}⁣}:{\mathcal{O}}_X\to g_{\ast }{\mathcal{O}}_{X^{\mathrm{\prime }}}\stackrel{g_{\ast }{\mathrm{d}}_{X^{\mathrm{\prime }}\mathrm{/}{S}^{\mathrm{\prime }}}}{\to }{g}_{\ast }{\mathrm{\Omega }}_{X^{\mathrm{\prime }}\mathrm{/}{S}^{\mathrm{\prime }}}$ is an $S$-derivation of ${\mathcal{O}}_X$ into the ${\mathcal{O}}_X$-module $g_{\ast }{\mathrm{\Omega }}_{X^{\mathrm{\prime }}\mathrm{/}{S}^{\mathrm{\prime }}}$: it is additive, satisfies Leibniz for the ${\mathcal{O}}_X$-module structure transported along $g^{\mathrm{♯}}$, and kills the image of ${\mathcal{O}}_S$, because that image is mapped into the image of the ${\mathcal{O}}_{S^{\mathrm{\prime }}}$-structure of $X^{\mathrm{\prime }}$, which ${\mathrm{d}}_{X^{\mathrm{\prime }}\mathrm{/}{S}^{\mathrm{\prime }}}$ annihilates by [F3]. By [F4] there is a unique ${\mathcal{O}}_X$-linear $u\text{\hspace{0.17em}⁣}:{\mathrm{\Omega }}_{X\mathrm{/}S}\to g_{\ast }{\mathrm{\Omega }}_{X^{\mathrm{\prime }}\mathrm{/}{S}^{\mathrm{\prime }}}$ with $u({\mathrm{d}}_{X\mathrm{/}S}(a))={\mathrm{d}}_{X^{\mathrm{\prime }}\mathrm{/}{S}^{\mathrm{\prime }}}(a\circ g)$, and by [F5] there is a unique ${\mathcal{O}}_{X^{\mathrm{\prime }}}$-linear map $\gamma \text{\hspace{0.17em}⁣}:g^{\ast }{\mathrm{\Omega }}_{X\mathrm{/}S}\to {\mathrm{\Omega }}_{X^{\mathrm{\prime }}\mathrm{/}{S}^{\mathrm{\prime }}}$ sending $1\otimes {\mathrm{d}}_{X\mathrm{/}S}(a)$ to ${\mathrm{d}}_{X^{\mathrm{\prime }}\mathrm{/}{S}^{\mathrm{\prime }}}(a\circ g)$.

Affine charts compute $\gamma$. Let $x^{\mathrm{\prime }}\in X^{\mathrm{\prime }}$ have image $x\in X$ and $s\in S$, and choose an affine open $V=\mathrm{Spec}A\subseteq S$ containing $s$; then choose an affine open $U=\mathrm{Spec}B\subseteq X$ containing $x$ with image in $V$ and an affine open $W=\mathrm{Spec}{A}^{\mathrm{\prime }}\subseteq S^{\mathrm{\prime }}$ containing the image of $x^{\mathrm{\prime }}$ with image in $V$. By [F2], $U^{\mathrm{\prime }}:=U{\times }_{V}W=\mathrm{Spec}{B}^{\mathrm{\prime }}$ for $B^{\mathrm{\prime }}=B{\otimes }_A{A}^{\mathrm{\prime }}$ is an open affine neighbourhood of $x^{\mathrm{\prime }}$. By [F7], ${\mathrm{\Omega }}_{X\mathrm{/}S}{\mathrm{\mid }}_U$ and ${\mathrm{\Omega }}_{X^{\mathrm{\prime }}\mathrm{/}{S}^{\mathrm{\prime }}}{\mathrm{\mid }}_{U^{\mathrm{\prime }}}$ are attached to ${\mathrm{\Omega }}_{B\mathrm{/}A}$ and ${\mathrm{\Omega }}_{B^{\mathrm{\prime }}\mathrm{/}{A}^{\mathrm{\prime }}}$. If $x^{\mathrm{\prime }}$ corresponds to ${\mathfrak{p}}^{\mathrm{\prime }}\subseteq B^{\mathrm{\prime }}$ and $x$ to $\mathfrak{p}={\mathfrak{p}}^{\mathrm{\prime }}\cap B$, the pullback definition [F6] and stalk construction [F7] give $(g^{\ast }{\mathrm{\Omega }}_{X\mathrm{/}S}{)}_{x^{\mathrm{\prime }}}\cong ({\mathrm{\Omega }}_{B\mathrm{/}A}{)}_{\mathfrak{p}}{\otimes }_{B_{\mathfrak{p}}}{B}_{{\mathfrak{p}}^{\mathrm{\prime }}}^{\mathrm{\prime }}\cong (B^{\mathrm{\prime }}{\otimes }_B{\mathrm{\Omega }}_{B\mathrm{/}A}{)}_{{\mathfrak{p}}^{\mathrm{\prime }}}$. Hence the pullback is the sheaf attached to $B^{\mathrm{\prime }}{\otimes }_B{\mathrm{\Omega }}_{B\mathrm{/}A}$ on $U^{\mathrm{\prime }}$. The map $\gamma$ sends $1\otimes \mathrm{d}b$ to $\mathrm{d}(b\otimes 1)$ by step 1.1, so on these stalks it is the localisation of the canonical isomorphism of [F1].

$\gamma$ is an isomorphism. Every point $x^{\mathrm{\prime }}\in X^{\mathrm{\prime }}$ lies in a chart $U^{\mathrm{\prime }}$ as in step 2.1, on which $\gamma {\mathrm{\mid }}_{U^{\mathrm{\prime }}}$ is the isomorphism of [F1]; a morphism of sheaves whose restriction to each member of an open cover is an isomorphism is an isomorphism (equivalently, its stalk maps are isomorphisms), and forming stalks of the sheaves attached to $B^{\mathrm{\prime }}$-modules at points of $U^{\mathrm{\prime }}$ is compatible with the identifications of step 2.1. Hence $\gamma$ is an isomorphism of ${\mathcal{O}}_{X^{\mathrm{\prime }}}$-modules, with the asserted description on generators.

Naturality and hypotheses. The map $\gamma$ was produced from the universal properties of [F4] and the adjunction [F5] applied to the given morphisms $S^{\mathrm{\prime }}\to S$ and $X\to S$; replacing the base-change data by a morphism of squares replaces $\gamma$ by the corresponding pullback of $\gamma$, and on affine charts this is the naturality statement of [F1]. Only the existence of the fibre product and the affine descriptions of $\mathrm{\Omega }$ were used, so no flatness, finiteness, separatedness or tor-independence hypothesis enters.