Transitivity sequence for schemes

The map $\gamma$. The composite $D\text{\hspace{0.17em}⁣}:{\mathcal{O}}_Y\to f_{\ast }{\mathcal{O}}_X\stackrel{f_{\ast }{\mathrm{d}}_{X\mathrm{/}S}}{\to }{f}_{\ast }{\mathrm{\Omega }}_{X\mathrm{/}S}$ is an $S$-derivation: it is additive, satisfies Leibniz for the ${\mathcal{O}}_Y$-module structure of $f_{\ast }{\mathrm{\Omega }}_{X\mathrm{/}S}$ transported along $f^{\mathrm{♯}}$, and kills the image of ${\mathcal{O}}_S$ because [F1] applied to $X\to S$ says that ${\mathrm{d}}_{X\mathrm{/}S}$ annihilates it. By [F2] applied to the $S$-scheme $Y$ there is a unique ${\mathcal{O}}_Y$-linear $u\text{\hspace{0.17em}⁣}:{\mathrm{\Omega }}_{Y\mathrm{/}S}\to f_{\ast }{\mathrm{\Omega }}_{X\mathrm{/}S}$ with $u({\mathrm{d}}_{Y\mathrm{/}S}(g))={\mathrm{d}}_{X\mathrm{/}S}(g\circ f)$, and by [F4] there is a unique ${\mathcal{O}}_X$-linear $\gamma \text{\hspace{0.17em}⁣}:f^{\ast }{\mathrm{\Omega }}_{Y\mathrm{/}S}\to {\mathrm{\Omega }}_{X\mathrm{/}S}$ with $\gamma (1\otimes {\mathrm{d}}_{Y\mathrm{/}S}(g))={\mathrm{d}}_{X\mathrm{/}S}(g\circ f)$, where $f^{\ast }{\mathrm{\Omega }}_{Y\mathrm{/}S}={\mathcal{O}}_X{\otimes }_{f^{-1}{\mathcal{O}}_Y}{f}^{-1}{\mathrm{\Omega }}_{Y\mathrm{/}S}$ is the pullback of [F3] and the elements $1\otimes s$ generate it; the value on $1\otimes s$ is $u(s)$ by the description of the adjunction.

The map $\delta$. The universal $Y$-derivation ${\mathrm{d}}_{X\mathrm{/}Y}\text{\hspace{0.17em}⁣}:{\mathcal{O}}_X\to {\mathrm{\Omega }}_{X\mathrm{/}Y}$ annihilates the image of ${\mathcal{O}}_Y$, hence also the image of ${\mathcal{O}}_S$ under ${\mathcal{O}}_S\to {\mathcal{O}}_Y\to {\mathcal{O}}_X$; so it is an $S$-derivation, and [F2] over $S$ gives a unique ${\mathcal{O}}_X$-linear $\delta \text{\hspace{0.17em}⁣}:{\mathrm{\Omega }}_{X\mathrm{/}S}\to {\mathrm{\Omega }}_{X\mathrm{/}Y}$ with $\delta ({\mathrm{d}}_{X\mathrm{/}S}(c))={\mathrm{d}}_{X\mathrm{/}Y}(c)$. It is surjective because the sections ${\mathrm{d}}_{X\mathrm{/}Y}(c)$ generate ${\mathrm{\Omega }}_{X\mathrm{/}Y}$ over ${\mathcal{O}}_X$ by [F1].

The composite vanishes. For a local section $g$ of ${\mathcal{O}}_Y$ one has $\delta (\gamma (1\otimes \mathrm{d}g))=\delta ({\mathrm{d}}_{X\mathrm{/}S}(g\circ f))={\mathrm{d}}_{X\mathrm{/}Y}(g\circ f)=0$, because $g\circ f$ is the image of a section of ${\mathcal{O}}_Y$; hence $\mathrm{im}\gamma \subseteq \ker \delta$.

Affine charts. Let $V=\mathrm{Spec}B\subseteq Y$ be an affine open whose image lies in $W=\mathrm{Spec}A\subseteq S$, and let $U=\mathrm{Spec}C\subseteq X$ be an affine open with $f(U)\subseteq V$. The structure maps give $A\to B\to C$. By [F6], ${\mathrm{\Omega }}_{X\mathrm{/}S}{\mathrm{\mid }}_U$ and ${\mathrm{\Omega }}_{X\mathrm{/}Y}{\mathrm{\mid }}_U$ are attached to ${\mathrm{\Omega }}_{C\mathrm{/}A}$ and ${\mathrm{\Omega }}_{C\mathrm{/}B}$. For $x\in U$, let $\mathfrak{p}\subseteq C$ correspond to $x$ and $\mathfrak{q}=\mathfrak{p}\cap B$ to $f(x)$. The pullback definition [F3] and stalk construction [F9] give $(f^{\ast }{\mathrm{\Omega }}_{Y\mathrm{/}S}{)}_x\cong {\mathrm{\Omega }}_{Y\mathrm{/}S,f(x)}{\otimes }_{{\mathcal{O}}_{Y,f(x)}}{\mathcal{O}}_{X,x}\cong ({\mathrm{\Omega }}_{B\mathrm{/}A}{)}_{\mathfrak{q}}{\otimes }_{B_{\mathfrak{q}}}{C}_{\mathfrak{p}}\cong (C{\otimes }_B{\mathrm{\Omega }}_{B\mathrm{/}A}{)}_{\mathfrak{p}}$. Consequently $f^{\ast }{\mathrm{\Omega }}_{Y\mathrm{/}S}{\mathrm{\mid }}_U$ is the sheaf attached to $C{\otimes }_B{\mathrm{\Omega }}_{B\mathrm{/}A}$. These stalk identifications use tensor products after taking inverse-image stalks; no equality between $f^{-1}{\mathcal{O}}_Y$ and ${\mathcal{O}}_X$ is needed. The maps $\gamma$ and $\delta$ become the maps of [F5] because their values on $\mathrm{d}b$ and ${\mathrm{d}}_{C\mathrm{/}A}(c)$ are those of steps 1.1 and 1.2.

Exactness at ${\mathrm{\Omega }}_{X\mathrm{/}S}$. Let $x\in X$ and take a chart as in step 2.2 with $x$ corresponding to a prime $\mathfrak{p}\subseteq C$. By step 2.2 the stalks of the three sheaves at $x$ are $(C{\otimes }_B{\mathrm{\Omega }}_{B\mathrm{/}A}){\otimes }_C{C}_{\mathfrak{p}}$, ${\mathrm{\Omega }}_{C\mathrm{/}A}{\otimes }_C{C}_{\mathfrak{p}}$ and ${\mathrm{\Omega }}_{C\mathrm{/}B}{\otimes }_C{C}_{\mathfrak{p}}$, and the stalk maps are the localizations at $\mathfrak{p}$ of the maps of [F5]. Applying $-{\otimes }_C{C}_{\mathfrak{p}}$ to the exact sequence [F5] and using [F8], the stalk sequence is exact at the middle term, so $\mathrm{im}{\gamma }_x=\ker {\delta }_x$. As $x$ was arbitrary, [F7] gives $\mathrm{im}\gamma =\ker \delta$ and the sequence of the statement is exact at ${\mathrm{\Omega }}_{X\mathrm{/}S}$; combined with step 1.2 and step 2.1 this is the asserted exactness.

Failure of injectivity of the first arrow. Let $k$ be a field of characteristic $\mathrm{\ne }2$, let $A=k$, $B=k[x]$, $C=k[x]\mathrm{/}(x^2)$, so that ${\mathrm{\Omega }}_{B\mathrm{/}A}$ is free on $\mathrm{d}x$ by [F10] and ${\mathrm{\Omega }}_{C\mathrm{/}A}$ is the cokernel of $I\mathrm{/}{I}^2\to C{\otimes }_B{\mathrm{\Omega }}_{B\mathrm{/}A}$ for $I=(x^2)$. By [F10] the module $C{\otimes }_B{\mathrm{\Omega }}_{B\mathrm{/}A}=C\mathrm{d}x$ has the two $k$-linearly independent elements $1\otimes \mathrm{d}x$ and $x\otimes \mathrm{d}x$, while $\mathrm{d}(x^2)=2x\mathrm{d}x$ shows that $x\otimes \mathrm{d}x$ lies in the kernel of $C{\otimes }_B{\mathrm{\Omega }}_{B\mathrm{/}A}\to {\mathrm{\Omega }}_{C\mathrm{/}A}$; since $2\mathrm{\ne }0$ in $k$, the element $1\otimes \mathrm{d}x$ does not, so this map has a nonzero kernel and $\gamma$ is not injective in general.

Conclusion. Steps 1.1 and 1.2 construct $\gamma$ and $\delta$ with the stated properties, step 2.1 shows that the composite vanishes, step 3.1 identifies the kernel of $\delta$ with the image of $\gamma$ and makes $\delta$ surjective by step 1.2, and step 3.2 shows that $\gamma$ need not be injective. Hence the displayed sequence is exact and the first arrow is not injective in general. Naturality in $f$ and $h$ follows because $\gamma$ and $\delta$ are determined by the universal properties of [F2] and [F4] applied to the morphisms $f$ and $h$, which are natural in those morphisms, and no finiteness, flatness or separatedness assumption was used.