Lemma 31.13.13. Let $f : X \to Y$ be a morphism of schemes. Let $D \subset Y$ be an effective Cartier divisor. The pullback of $D$ by $f$ is defined in each of the following cases:

$f(x) \not\in D$ for any weakly associated point $x$ of $X$,

$X$, $Y$ integral and $f$ dominant,

$X$ reduced and $f(\xi ) \not\in D$ for any generic point $\xi $ of any irreducible component of $X$,

$X$ is locally Noetherian and $f(x) \not\in D$ for any associated point $x$ of $X$,

$X$ is locally Noetherian, has no embedded points, and $f(\xi ) \not\in D$ for any generic point $\xi $ of an irreducible component of $X$,

$f$ is flat, and

add more here as needed.

## Comments (0)

There are also: