Let $X \to S$ be a morphism of schemes. Let $Y \subset X$ be an effective Cartier divisor. If $X$ étale locally along $Y$ looks like $Y \times \mathbf{A}^1$, then there is a canonical short exact sequence of complexes
having many good properties we will discuss in this section. There is a variant of this construction where one starts with a normal crossings divisor (Étale Morphisms, Definition 41.21.1) which we will discuss elsewhere (insert future reference here).
Definition 50.15.1. Let $X \to S$ be a morphism of schemes. Let $Y \subset X$ be an effective Cartier divisor. We say the de Rham complex of log poles is defined for $Y \subset X$ over $S$ if for all $y \in Y$ and local equation $f \in \mathcal{O}_{X, y}$ of $Y$ we have
$\mathcal{O}_{X, y} \to \Omega _{X/S, y}$, $g \mapsto g \text{d}f$ is a split injection, and
$\Omega ^ p_{X/S, y}$ is $f$-torsion free for all $p$.
An easy local calculation shows that it suffices for every $y \in Y$ to find one local equation $f$ for which conditions (1) and (2) hold.
Lemma 50.15.2. Let $X \to S$ 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 $S$. There is a canonical short exact sequence of complexes
\[ 0 \to \Omega ^\bullet _{X/S} \to \Omega ^\bullet _{X/S}(\log Y) \to \Omega ^\bullet _{Y/S}[-1] \to 0 \]
Proof. Our assumption is that for every $y \in Y$ and local equation $f \in \mathcal{O}_{X, y}$ of $Y$ we have
for some module $M$ with $f$-torsion free exterior powers $\wedge ^ p(M)$. It follows that
Below we will tacitly use these facts. In particular the sheaves $\Omega ^ p_{X/S}$ have no nonzero local sections supported on $Y$ and we have a canonical inclusion
see More on Flatness, Section 38.42. Let $U = \mathop{\mathrm{Spec}}(A)$ be an affine open subscheme such that $Y \cap U = V(f)$ for some nonzerodivisor $f \in A$. Let us consider the $\mathcal{O}_ U$-submodule of $\Omega ^ p_{X/S}(Y)|_ U$ generated by $\Omega ^ p_{X/S}|_ U$ and $\text{d}\log (f) \wedge \Omega ^{p - 1}_{X/S}$ where $\text{d}\log (f) = f^{-1}\text{d}(f)$. This is independent of the choice of $f$ as another generator of the ideal of $Y$ on $U$ is equal to $uf$ for a unit $u \in A$ and we get
which is a section of $\Omega _{X/S}$ over $U$. These local sheaves glue to give a quasi-coherent submodule
Let us agree to think of $\Omega ^ p_{Y/S}$ as a quasi-coherent $\mathcal{O}_ X$-module. There is a unique surjective $\mathcal{O}_ X$-linear map
defined by the rule
for all opens $U$ as above and all $\eta ' \in \Omega ^ p_{X/S}(U)$ and $\eta \in \Omega ^{p - 1}_{X/S}(U)$. If a form $\eta $ over $U$ restricts to zero on $Y \cap U$, then $\eta = \text{d}f \wedge \eta ' + f\eta ''$ for some forms $\eta '$ and $\eta ''$ over $U$. We conclude that we have a short exact sequence
for all $p$. We still have to define the differentials $\Omega ^ p_{X/S}(\log Y) \to \Omega ^{p + 1}_{X/S}(\log Y)$. On the subsheaf $\Omega ^ p_{X/S}$ we use the differential of the de Rham complex of $X$ over $S$. Finally, we define $\text{d}(\text{d}\log (f) \wedge \eta ) = -\text{d}\log (f) \wedge \text{d}\eta $. The sign is forced on us by the Leibniz rule (on $\Omega ^\bullet _{X/S}$) and it is compatible with the differential on $\Omega ^\bullet _{Y/S}[-1]$ which is after all $-\text{d}_{Y/S}$ by our sign convention in Homology, Definition 12.14.7. In this way we obtain a short exact sequence of complexes as stated in the lemma. $\square$
Definition 50.15.3. Let $X \to S$ 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 $S$. Then the complex constructed in Lemma 50.15.2 is the de Rham complex of log poles for $Y \subset X$ over $S$.
\[ \Omega ^\bullet _{X/S}(\log Y) \]
This complex has many good properties.
Lemma 50.15.4. Let $p : X \to S$ 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 $S$.
The maps $\wedge : \Omega ^ p_{X/S} \times \Omega ^ q_{X/S} \to \Omega ^{p + q}_{X/S}$ extend uniquely to $\mathcal{O}_ X$-bilinear maps
satisfying the Leibniz rule $ \text{d}(\omega \wedge \eta ) = \text{d}(\omega ) \wedge \eta + (-1)^{\deg (\omega )} \omega \wedge \text{d}(\eta )$,
with multiplication as in (1) the map $\Omega ^\bullet _{X/S} \to \Omega ^\bullet _{X/S}(\log (Y)$ is a homomorphism of differential graded $\mathcal{O}_ S$-algebras,
via the maps in (1) we have $\Omega ^ p_{X/S}(\log Y) = \wedge ^ p(\Omega ^1_{X/S}(\log Y))$, and
the map $\text{Res} : \Omega ^\bullet _{X/S}(\log Y) \to \Omega ^\bullet _{Y/S}[-1]$ satisfies
for $\omega $ a local section of $\Omega ^ p_{X/S}(\log Y)$ and $\eta $ a local section of $\Omega ^ q_{X/S}$.
\[ \wedge : \Omega ^ p_{X/S}(\log Y) \times \Omega ^ q_{X/S}(\log Y) \to \Omega ^{p + q}_{X/S}(\log Y) \]
\[ \text{Res}(\omega \wedge \eta ) = \text{Res}(\omega ) \wedge \eta |_ Y \]
Proof. This follows by direct calcuation from the local construction of the complex in the proof of Lemma 50.15.2. Details omitted. $\square$
Consider a commutative diagram
of schemes. Let $Y \subset X$ be an effective Cartier divisor whose pullback $Y' = f^*Y$ is defined (Divisors, Definition 31.13.12). Assume the de Rham complex of log poles is defined for $Y \subset X$ over $S$ and the de Rham complex of log poles is defined for $Y' \subset X'$ over $S'$. In this case we obtain a map of short exact sequences of complexes
Linearizing, for every $p$ we obtain a linear map $f^*\Omega ^ p_{X/S}(\log Y) \to \Omega ^ p_{X'/S'}(\log Y')$.
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 in $D(X, f^{-1}\mathcal{O}_ S)$ the “boundary” map coming from the short exact sequence in Lemma 50.15.2. Denote 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 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 is commutative in $D(X, f^{-1}\mathcal{O}_ S)$.
\[ \delta : \Omega ^\bullet _{Y/S} \to \Omega ^\bullet _{X/S}[2] \]
\[ \xi ' : \Omega ^\bullet _{X/S} \to \Omega ^\bullet _{X/S}[2] \]
\[ \zeta ' : \Omega ^\bullet _{Y/S} \to \Omega ^\bullet _{Y/S}[2] \]
\[ \xymatrix{ \Omega ^\bullet _{X/S} \ar[d]_{\xi '} \ar[r] & \Omega ^\bullet _{Y/S} \ar[d]^{\zeta '} \ar[ld]_\delta \\ \Omega ^\bullet _{X/S}[2] \ar[r] & \Omega ^\bullet _{Y/S}[2] } \]
Proof. More precisely, we define $\delta $ as the boundary map corresponding to the shifted short exact sequence
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
where the left and right vertical arrows are the obvious ones. We can define the middle vertical arrow by the rule
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
where the left and right vertical arrows are the obvious ones. We can define the middle vertical arrow by the rule
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. $\square$
Lemma 50.15.6. Let $X \to S$ 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 $S$. Let $b \in H^ m_{dR}(X/S)$ be a de Rham cohomology class whose restriction to $Y$ is zero. Then $c_1^{dR}(\mathcal{O}_ X(Y)) \cup b = 0$ in $H^{m + 2}_{dR}(X/S)$.
Proof. This follows immediately from Lemma 50.15.5. Namely, we have
as desired. For the second equality, see Remark 50.4.3. $\square$
Lemma 50.15.7. Let $X \to T \to S$ be morphisms of schemes. Let $Y \subset X$ be an effective Cartier divisor. If both $X \to T$ and $Y \to T$ are smooth, then the de Rham complex of log poles is defined for $Y \subset X$ over $S$.
Proof. Let $y \in Y$ be a point. By More on Morphisms, Lemma 37.17.1 there exists an integer $0 \geq m$ and a commutative diagram
where $U \subset X$ is open, $V = Y \cap U$, $\pi $ is étale, $V = \pi ^{-1}(\mathbf{A}^ m_ T)$, and $y \in V$. Denote $z \in \mathbf{A}^ m_ T$ the image of $y$. Then we have
by Lemma 50.2.2. Denote $x_1, \ldots , x_{m + 1}$ the coordinate functions on $\mathbf{A}^{m + 1}_ T$. Since the conditions (1) and (2) in Definition 50.15.1 do not depend on the choice of the local coordinate, it suffices to check the conditions (1) and (2) when $f$ is the image of $x_{m + 1}$ by the flat local ring homomorphism $\mathcal{O}_{\mathbf{A}^{m + 1}_ T, z} \to \mathcal{O}_{X, x}$. In this way we see that it suffices to check conditions (1) and (2) for $\mathbf{A}^ m_ T \subset \mathbf{A}^{m + 1}_ T$ and the point $z$. To prove this case we may assume $S = \mathop{\mathrm{Spec}}(A)$ and $T = \mathop{\mathrm{Spec}}(B)$ are affine. Let $A \to B$ be the ring map corresponding to the morphism $T \to S$ and set $P = B[x_1, \ldots , x_{m + 1}]$ so that $\mathbf{A}^{m + 1}_ T = \mathop{\mathrm{Spec}}(B)$. We have
Hence the map $P \to \Omega _{P/A}$, $g \mapsto g \text{d}x_{m + 1}$ is a split injection and $x_{m + 1}$ is a nonzerodivisor on $\Omega ^ p_{P/A}$ for all $p \geq 0$. Localizing at the prime ideal corresponding to $z$ finishes the proof. $\square$
Remark 50.15.8. Let $S$ be a locally Noetherian scheme. Let $X$ be locally of finite type over $S$. Let $Y \subset X$ be an effective Cartier divisor. If the map has a section for all $y \in Y$, then the de Rham complex of log poles is defined for $Y \subset X$ over $S$. If we ever need this result we will formulate a precise statement and add a proof here.
\[ \mathcal{O}_{X, y}^\wedge \longrightarrow \mathcal{O}_{Y, y}^\wedge \]
Remark 50.15.9. Let $S$ be a locally Noetherian scheme. Let $X$ be locally of finite type over $S$. Let $Y \subset X$ be an effective Cartier divisor. If for every $y \in Y$ we can find a diagram of schemes over $S$ with $\varphi $ étale and $\psi |_{\varphi ^{-1}(Y)} : \varphi ^{-1}(Y) \to V$ étale, then the de Rham complex of log poles is defined for $Y \subset X$ over $S$. A special case is when the pair $(X, Y)$ étale locally looks like $(V \times \mathbf{A}^1, V \times \{ 0\} )$. If we ever need this result we will formulate a precise statement and add a proof here.
\[ X \xleftarrow {\varphi } U \xrightarrow {\psi } V \]
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