The Stacks project

Lemma 31.7.3. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $g : S' \to S$ be a morphism of schemes. Consider the base change diagram

\[ \xymatrix{ X' \ar[d] \ar[r]_{g'} & X \ar[d] \\ S' \ar[r]^ g & S } \]

and set $\mathcal{F}' = (g')^*\mathcal{F}$. Let $x' \in X'$ be a point with images $x \in X$, $s' \in S'$ and $s \in S$. Assume $f$ locally of finite type. Then $x' \in \text{Ass}_{X'/S'}(\mathcal{F}')$ if and only if $x \in \text{Ass}_{X/S}(\mathcal{F})$ and $x'$ corresponds to a generic point of an irreducible component of $\mathop{\mathrm{Spec}}(\kappa (s') \otimes _{\kappa (s)} \kappa (x))$.

Proof. Consider the morphism $X'_{s'} \to X_ s$ of fibres. As $X_{s'} = X_ s \times _{\mathop{\mathrm{Spec}}(\kappa (s))} \mathop{\mathrm{Spec}}(\kappa (s'))$ this is a flat morphism. Moreover $\mathcal{F}'_{s'}$ is the pullback of $\mathcal{F}_ s$ via this morphism. As $X_ s$ is locally of finite type over the Noetherian scheme $\mathop{\mathrm{Spec}}(\kappa (s))$ we have that $X_ s$ is locally Noetherian, see Morphisms, Lemma 29.15.6. Thus we may apply Lemma 31.3.1 and we see that

\[ \text{Ass}_{X'_{s'}}(\mathcal{F}'_{s'}) = \bigcup \nolimits _{x \in \text{Ass}(\mathcal{F}_ s)} \text{Ass}((X'_{s'})_ x). \]

Thus to prove the lemma it suffices to show that the associated points of the fibre $(X'_{s'})_ x$ of the morphism $X'_{s'} \to X_ s$ over $x$ are its generic points. Note that $(X'_{s'})_ x = \mathop{\mathrm{Spec}}(\kappa (s') \otimes _{\kappa (s)} \kappa (x))$ as schemes. By Algebra, Lemma 10.167.1 the ring $\kappa (s') \otimes _{\kappa (s)} \kappa (x)$ is a Noetherian Cohen-Macaulay ring. Hence its associated primes are its minimal primes, see Algebra, Proposition 10.63.6 (minimal primes are associated) and Algebra, Lemma 10.157.2 (no embedded primes). $\square$

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