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 $S$.

1. 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

$\wedge : \Omega ^ p_{X/S}(\log Y) \times \Omega ^ q_{X/S}(\log Y) \to \Omega ^{p + q}_{X/S}(\log Y)$

satisfying the Leibniz rule $\text{d}(\omega \wedge \eta ) = \text{d}(\omega ) \wedge \eta + (-1)^{\deg (\omega )} \omega \wedge \text{d}(\eta )$,

2. 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,

3. via the maps in (1) we have $\Omega ^ p_{X/S}(\log Y) = \wedge ^ p(\Omega ^1_{X/S}(\log Y))$, and

4. the map $\text{Res} : \Omega ^\bullet _{X/S}(\log Y) \to \Omega ^\bullet _{Y/S}[-1]$ satisfies

$\text{Res}(\omega \wedge \eta ) = \text{Res}(\omega ) \wedge \eta |_ Y$

for $\omega$ a local section of $\Omega ^ p_{X/S}(\log Y)$ and $\eta$ a local section of $\Omega ^ q_{X/S}$.

Proof. This follows by direct calcuation from the local construction of the complex in the proof of Lemma 50.15.2. Details omitted. $\square$

Comment #6218 by Martin Bright on

In (4), has $\omega|Y$ been defined for $\omega \in \Omega^p_{X/S}(\log Y)$?

Comment #6220 by on

Hmm... yes, that seems bad. I will fix this the next time I go through all the comments,

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).