Conormal sequence for a closed immersion

The map $\alpha$. For a local section $t$ of $\mathcal{I}$ over an open $V\subseteq Y$ let $\alpha (t)\in (i^{\ast }{\mathrm{\Omega }}_{Y\mathrm{/}S})(i^{-1}V)$ be the image of ${\mathrm{d}}_{Y\mathrm{/}S}(t)$ under the canonical map $i^{-1}{\mathrm{\Omega }}_{Y\mathrm{/}S}\to i^{\ast }{\mathrm{\Omega }}_{Y\mathrm{/}S}$ of [F2], i.e. $1\otimes {\mathrm{d}}_{Y\mathrm{/}S}(t)$. For a local section $g$ of ${\mathcal{O}}_Y$ over $V$ one has $\mathrm{d}(gt)=g\mathrm{d}t+t\mathrm{d}g$, hence $1\otimes \mathrm{d}(gt)=(g\circ i)(1\otimes \mathrm{d}t)+(t\circ i)(1\otimes \mathrm{d}g)=(g\circ i)(1\otimes \mathrm{d}t)$ because $t$ lies in the kernel of ${\mathcal{O}}_Y\to i_{\ast }{\mathcal{O}}_X$, so that $t\circ i=0$; thus these formulas define an ${\mathcal{O}}_Y$-linear map $\mathcal{I}\to i_{\ast }{i}^{\ast }{\mathrm{\Omega }}_{Y\mathrm{/}S}$. For local sections $t,t^{\mathrm{\prime }}$ of $\mathcal{I}$ one has $1\otimes \mathrm{d}(t{t}^{\mathrm{\prime }})=(t\circ i)(1\otimes \mathrm{d}{t}^{\mathrm{\prime }})+(t^{\mathrm{\prime }}\circ i)(1\otimes \mathrm{d}t)=0$, so $\alpha$ kills ${\mathcal{I}}^2$, and since $\mathcal{I}$ annihilates both $\mathcal{I}\mathrm{/}{\mathcal{I}}^2$ and the pullback (a local section $t$ of $\mathcal{I}$ acts on $i^{\ast }{\mathrm{\Omega }}_{Y\mathrm{/}S}$ as $t\circ i=0$), the descended formulas on germs define a map $i^{-1}\mathcal{Q}\to i^{\ast }{\mathrm{\Omega }}_{Y\mathrm{/}S}$. It is linear over $i^{-1}{\mathcal{O}}_Y$ and hence over its quotient ${\mathcal{O}}_X$, so this is the required ${\mathcal{O}}_X$-linear $\alpha$.

The map $\beta$. Since $Y$ and $X$ are $S$-schemes and $i$ is an $S$-morphism, the composite $D\text{\hspace{0.17em}⁣}:{\mathcal{O}}_Y\to i_{\ast }{\mathcal{O}}_X\to i_{\ast }{\mathrm{\Omega }}_{X\mathrm{/}S}$ of the structure map $i^{\mathrm{♯}}$ with $i_{\ast }{\mathrm{d}}_{X\mathrm{/}S}$ is additive, satisfies Leibniz for the ${\mathcal{O}}_Y$-module structure of $i_{\ast }{\mathrm{\Omega }}_{X\mathrm{/}S}$ transported along $i^{\mathrm{♯}}$, and kills the image of ${\mathcal{O}}_S$: the image of $g^{-1}{\mathcal{O}}_S\to {\mathcal{O}}_Y\to i_{\ast }{\mathcal{O}}_X$ is the image of $f^{-1}{\mathcal{O}}_S\to {\mathcal{O}}_X$ for the structure morphism $f\text{\hspace{0.17em}⁣}:X\to S$, which ${\mathrm{d}}_{X\mathrm{/}S}$ annihilates. Hence $D$ is an $S$-derivation of ${\mathcal{O}}_Y$ into $i_{\ast }{\mathrm{\Omega }}_{X\mathrm{/}S}$, and [F7] applied to the $S$-scheme $Y$ gives a unique ${\mathcal{O}}_Y$-linear map $u\text{\hspace{0.17em}⁣}:{\mathrm{\Omega }}_{Y\mathrm{/}S}\to i_{\ast }{\mathrm{\Omega }}_{X\mathrm{/}S}$ with $u({\mathrm{d}}_{Y\mathrm{/}S}(g))={\mathrm{d}}_{X\mathrm{/}S}(g\circ i)$. Let $\beta \text{\hspace{0.17em}⁣}:i^{\ast }{\mathrm{\Omega }}_{Y\mathrm{/}S}\to {\mathrm{\Omega }}_{X\mathrm{/}S}$ be the ${\mathcal{O}}_X$-linear map corresponding to $u$ under the adjunction [F8]; it satisfies $\beta (1\otimes {\mathrm{d}}_{Y\mathrm{/}S}(g))={\mathrm{d}}_{X\mathrm{/}S}(g\circ i)$ by the description of the correspondence, and it is unique with this property.

The composite vanishes. For a local section $t$ of $\mathcal{I}$ one has $\beta (\alpha (t))=\beta (1\otimes \mathrm{d}t)={\mathrm{d}}_{X\mathrm{/}S}(t\circ i)={\mathrm{d}}_{X\mathrm{/}S}(0)=0$ by the characterisations of steps 1.1 and 1.2, so the image of $\alpha$ is contained in the kernel of $\beta$.

Affine charts. Let $V=\mathrm{Spec}P\subseteq Y$ be an affine open with image in an affine open $W=\mathrm{Spec}A\subseteq S$, and take $U=i^{-1}(V)$. Since $i$ is a closed immersion, [F3] identifies $U$ with $\mathrm{Spec}B$ for $B=P\mathrm{/}I$, where $I=\mathcal{I}(V)$, and $U\to V$ with the quotient morphism. By [F4], ${\mathrm{\Omega }}_{X\mathrm{/}S}{\mathrm{\mid }}_U$ and ${\mathrm{\Omega }}_{Y\mathrm{/}S}{\mathrm{\mid }}_V$ are attached to ${\mathrm{\Omega }}_{B\mathrm{/}A}$ and ${\mathrm{\Omega }}_{P\mathrm{/}A}$. The ideal sheaf $\mathcal{I}{\mathrm{\mid }}_V$ is attached to $I$ by [F3]; on every principal open $D(h)\subseteq V$, multiplication has image $(I^2{)}_h$, so ${\mathcal{I}}^2{\mathrm{\mid }}_V$ is attached to $I^2$ and $\mathcal{I}\mathrm{/}{\mathcal{I}}^2$ on $U$ is attached to $I\mathrm{/}{I}^2$. For the pullback, let $x\in U$ correspond to $\mathfrak{p}\subseteq B$ and $i(x)$ to $\mathfrak{q}\subseteq P$. By [F2] and the stalk construction [F11], $(i^{\ast }{\mathrm{\Omega }}_{Y\mathrm{/}S}{)}_x={\mathrm{\Omega }}_{Y\mathrm{/}S,i(x)}{\otimes }_{{\mathcal{O}}_{Y,i(x)}}{\mathcal{O}}_{X,x}\cong ({\mathrm{\Omega }}_{P\mathrm{/}A}{)}_{\mathfrak{q}}{\otimes }_{P_{\mathfrak{q}}}{B}_{\mathfrak{p}}\cong (B{\otimes }_P{\mathrm{\Omega }}_{P\mathrm{/}A}{)}_{\mathfrak{p}}$. The last equality follows by localising the tensor product; it does not identify $i^{-1}{\mathcal{O}}_V$ with ${\mathcal{O}}_U$. Thus the pullback is the sheaf attached to $B{\otimes }_P{\mathrm{\Omega }}_{P\mathrm{/}A}$. On these stalks the maps of steps 1.1 and 1.2 agree with [F5]: $\alpha$ sends $[t]$ to $1\otimes \mathrm{d}t$ and $\beta$ sends $1\otimes \mathrm{d}g$ to $\mathrm{d}(g+I)$.

Surjectivity of $\beta$. By [F6] surjectivity of a morphism of sheaves may be checked on stalks. Every point of $X$ lies in a chart as in step 2.2, and on that chart $\beta$ becomes the second map of the exact sequence [F5], which is surjective; forming the stalk at a point of the chart is a filtered colimit of localizations and preserves surjectivity. Hence $\beta$ is surjective.

Exactness at the middle term. Let $x\in X$, choose a chart $U=\mathrm{Spec}B$ and $V=\mathrm{Spec}P$ as in step 2.2 with $x$ corresponding to a prime $\mathfrak{p}\subseteq B$ and $i(x)$ to $\mathfrak{p}\cap P\supseteq I$. By step 2.2 the stalks of the three sheaves at $x$ are $(I\mathrm{/}{I}^2){\otimes }_B{B}_{\mathfrak{p}}$, $(B{\otimes }_P{\mathrm{\Omega }}_{P\mathrm{/}A}){\otimes }_B{B}_{\mathfrak{p}}$ and ${\mathrm{\Omega }}_{B\mathrm{/}A}{\otimes }_B{B}_{\mathfrak{p}}$, and the stalk maps are the localizations of the maps of [F5] at $\mathfrak{p}$. Applying $-{\otimes }_B{B}_{\mathfrak{p}}$ to the exact sequence [F5] and using [F9], the stalk sequence is exact at the middle term, so $\mathrm{im}{\alpha }_x=\ker {\beta }_x$. Since $x$ was arbitrary, [F6] gives $\mathrm{im}\alpha =\ker \beta$ as subsheaves of $i^{\ast }{\mathrm{\Omega }}_{Y\mathrm{/}S}$.

Failure of injectivity. Take $A=k$ a field, $P=k[x]$, $I=(x^2)$ and $B=k[x]\mathrm{/}(x^2)$, so that $I\mathrm{/}{I}^2=(x^2)\mathrm{/}(x^4)$, in which the class of $x^3$ is nonzero. By [F10] we have $B{\otimes }_P{\mathrm{\Omega }}_{P\mathrm{/}A}=B\mathrm{d}x$ with $\mathrm{d}x$ a free generator, and $\alpha ([x^3])=1\otimes \mathrm{d}(x^3)=3x^2(1\otimes \mathrm{d}x)=0$, because $x^2=0$ in $B$: the class of $x^3$ lies in the kernel of $\alpha$ and is nonzero. Hence the left map of the conormal sequence is not injective in general, and in particular no injectivity is claimed.

Conclusion. Steps 1.1 and 1.2 construct ${\mathcal{O}}_X$-linear maps $\alpha$ and $\beta$ with the asserted descriptions, step 2.1 shows $\beta \circ \alpha =0$, step 3.1 shows that $\beta$ is surjective and step 3.2 that its kernel is exactly the image of $\alpha$; step 3.3 exhibits a case where $\alpha$ has nonzero kernel. Hence the displayed sequence of ${\mathcal{O}}_X$-modules is exact and its left map is not generally injective, with no finiteness, flatness or separatedness hypothesis used anywhere.