The Stacks project

Remark 31.7.4. With notation and assumptions as in Lemma 31.7.3 we see that it is always the case that $(g')^{-1}(\text{Ass}_{X/S}(\mathcal{F})) \supset \text{Ass}_{X'/S'}(\mathcal{F}')$. If the morphism $S' \to S$ is locally quasi-finite, then we actually have

\[ (g')^{-1}(\text{Ass}_{X/S}(\mathcal{F})) = \text{Ass}_{X'/S'}(\mathcal{F}') \]

because in this case the field extensions $\kappa (s) \subset \kappa (s')$ are always finite. In fact, this holds more generally for any morphism $g : S' \to S$ such that all the field extensions $\kappa (s) \subset \kappa (s')$ are algebraic, because in this case all prime ideals of $\kappa (s') \otimes _{\kappa (s)} \kappa (x)$ are maximal (and minimal) primes, see Algebra, Lemma 10.35.19.


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