Definition 50.15.1 (0FMV)—The Stacks project

Table of contents
Part 3: Topics in Scheme Theory
Chapter 50: de Rham Cohomology
Section 50.15: Log poles along a divisor
Definition 50.15.1 (cite)

next lemma

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 $p$.

Comments (3)

Comment #9052 by ZL on May 05, 2024 at 12:46

Typo: " $\Omega ^ p_{X/S, y}$ is $f$ -torsion free for all $p$ " should be " $\Omega ^ p_{X/S, y}$ is $f$ -torsion free for all $y$ ".

Comment #9179 by Stacks project on May 14, 2024 at 09:44

No! The sentence already says "for all $y$ in $Y$ " and we really want this to be true for all $p$ . Maybe I misunderstood your comment?

Comment #9314 by ZL on May 30, 2024 at 04:54

Thanks for the clarification. I was being silly!

There are also:

1 comment(s) on Section 50.15: Log poles along a divisor

Comments (3)

Post a comment