Lemma 50.15.5. Let $f : X \to S$ be a morphism of schemes. Let $Y \subset X$ be an effective Cartier divisor. Assume the de Rham complex of log poles is defined for $Y \subset X$ over $S$. Denote
\[ \delta : \Omega ^\bullet _{Y/S} \to \Omega ^\bullet _{X/S}[2] \]
in $D(X, f^{-1}\mathcal{O}_ S)$ the “boundary” map coming from the short exact sequence in Lemma 50.15.2. Denote
\[ \xi ' : \Omega ^\bullet _{X/S} \to \Omega ^\bullet _{X/S}[2] \]
in $D(X, f^{-1}\mathcal{O}_ S)$ the map of Remark 50.4.3 corresponding to $\xi = c_1^{dR}(\mathcal{O}_ X(-Y))$. Denote
\[ \zeta ' : \Omega ^\bullet _{Y/S} \to \Omega ^\bullet _{Y/S}[2] \]
in $D(Y, f|_ Y^{-1}\mathcal{O}_ S)$ the map of Remark 50.4.3 corresponding to $\zeta = c_1^{dR}(\mathcal{O}_ X(-Y)|_ Y)$. Then the diagram
\[ \xymatrix{ \Omega ^\bullet _{X/S} \ar[d]_{\xi '} \ar[r] & \Omega ^\bullet _{Y/S} \ar[d]^{\zeta '} \ar[ld]_\delta \\ \Omega ^\bullet _{X/S}[2] \ar[r] & \Omega ^\bullet _{Y/S}[2] } \]
is commutative in $D(X, f^{-1}\mathcal{O}_ S)$.
Proof.
More precisely, we define $\delta $ as the boundary map corresponding to the shifted short exact sequence
\[ 0 \to \Omega ^\bullet _{X/S}[1] \to \Omega ^\bullet _{X/S}(\log Y)[1] \to \Omega ^\bullet _{Y/S} \to 0 \]
It suffices to prove each triangle commutes. Set $\mathcal{L} = \mathcal{O}_ X(-Y)$. Denote $\pi : L \to X$ the line bundle with $\pi _*\mathcal{O}_ L = \bigoplus _{n \geq 0} \mathcal{L}^{\otimes n}$.
Commutativity of the upper left triangle. By Lemma 50.10.3 the map $\xi '$ is the boundary map of the triangle given in Lemma 50.10.2. By functoriality it suffices to prove there exists a morphism of short exact sequences
\[ \xymatrix{ 0 \ar[r] & \Omega ^\bullet _{X/S}[1] \ar[r] \ar[d] & \Omega ^\bullet _{L^\star /S, 0}[1] \ar[r] \ar[d] & \Omega ^\bullet _{X/S} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \Omega ^\bullet _{X/S}[1] \ar[r] & \Omega ^\bullet _{X/S}(\log Y)[1] \ar[r] & \Omega ^\bullet _{Y/S} \ar[r] & 0 } \]
where the left and right vertical arrows are the obvious ones. We can define the middle vertical arrow by the rule
\[ \omega ' + \text{d}\log (s) \wedge \omega \longmapsto \omega ' + \text{d}\log (f) \wedge \omega \]
where $\omega ', \omega $ are local sections of $\Omega ^\bullet _{X/S}$ and where $s$ is a local generator of $\mathcal{L}$ and $f \in \mathcal{O}_ X(-Y)$ is the corresponding section of the ideal sheaf of $Y$ in $X$. Since the constructions of the maps in Lemmas 50.10.2 and 50.15.2 match exactly, this works.
Commutativity of the lower right triangle. Denote $\overline{L}$ the restriction of $L$ to $Y$. By Lemma 50.10.3 the map $\zeta '$ is the boundary map of the triangle given in Lemma 50.10.2 using the line bundle $\overline{L}$ on $Y$. By functoriality it suffices to prove there exists a morphism of short exact sequences
\[ \xymatrix{ 0 \ar[r] & \Omega ^\bullet _{X/S}[1] \ar[r] \ar[d] & \Omega ^\bullet _{X/S}(\log Y)[1] \ar[r] \ar[d] & \Omega ^\bullet _{Y/S} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \Omega ^\bullet _{Y/S}[1] \ar[r] & \Omega ^\bullet _{\overline{L}^\star /S, 0}[1] \ar[r] & \Omega ^\bullet _{Y/S} \ar[r] & 0 \\ } \]
where the left and right vertical arrows are the obvious ones. We can define the middle vertical arrow by the rule
\[ \omega ' + \text{d}\log (f) \wedge \omega \longmapsto \omega '|_ Y + \text{d}\log (\overline{s}) \wedge \omega |_ Y \]
where $\omega ', \omega $ are local sections of $\Omega ^\bullet _{X/S}$ and where $f$ is a local generator of $\mathcal{O}_ X(-Y)$ viewed as a function on $X$ and where $\overline{s}$ is $f|_ Y$ viewed as a section of $\mathcal{L}|_ Y = \mathcal{O}_ X(-Y)|_ Y$. Since the constructions of the maps in Lemmas 50.10.2 and 50.15.2 match exactly, this works.
$\square$
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