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The Stacks project

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:

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

  2. X, Y integral and f dominant,

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

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

  5. 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,

  6. f is flat, and

  7. add more here as needed.

Proof. The question is local on X, and hence we reduce to the case where X = \mathop{\mathrm{Spec}}(A), Y = \mathop{\mathrm{Spec}}(R), f is given by \varphi : R \to A and D = \mathop{\mathrm{Spec}}(R/(t)) where t \in R is a nonzerodivisor. The goal in each case is to show that \varphi (t) \in A is a nonzerodivisor.

In case (1) this follows from Algebra, Lemma 10.66.7. Case (4) is a special case of (1) by Lemma 31.5.8. Case (5) follows from (4) and the definitions. Case (3) is a special case of (1) by Lemma 31.5.12. Case (2) is a special case of (3). If R \to A is flat, then t : R \to R being injective shows that t : A \to A is injective. This proves (6). \square


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