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 $p$.

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