The Stacks project

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 $\Omega ^ q_{X/S}$.