Lemma 50.15.2 (0FMW)—The Stacks project

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

$\Omega _{X/S, y} = \mathcal{O}_{X, y}\text{d}f \oplus M \quad \text{and}\quad \Omega ^ p_{X/S, y} = \wedge ^{p - 1}(M)\text{d}f \oplus \wedge ^ p(M)$

for some module $M$ with $f$ -torsion free exterior powers $\wedge ^ p(M)$ . It follows that

$\Omega ^ p_{Y/S, y} = \wedge ^ p(M/fM) = \wedge ^ p(M)/f\wedge ^ p(M)$

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

$\Omega ^ p_{X/S} \subset \Omega ^ p_{X/S}(Y)$

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

$\text{d}\log (uf) - \text{d}\log (f) = \text{d}\log (u) = u^{-1}\text{d}u$

which is a section of $\Omega _{X/S}$ over $U$ . These local sheaves glue to give a quasi-coherent submodule

$\Omega ^ p_{X/S} \subset \Omega ^ p_{X/S}(\log Y) \subset \Omega ^ p_{X/S}(Y)$

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

$\text{Res} : \Omega ^ p_{X/S}(\log Y) \to \Omega ^{p - 1}_{Y/S}$

defined by the rule

$\text{Res}(\eta ' + \text{d}\log (f) \wedge \eta ) = \eta |_{Y \cap U}$

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

$0 \to \Omega ^ p_{X/S} \to \Omega ^ p_{X/S}(\log Y) \to \Omega ^{p - 1}_{Y/S} \to 0$

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. $\square$

The Stacks project

Comments (0)

Post a comment