Section 31.8 (05AU): Relative weak assassin

31.8 Relative weak assassin

Definition 31.8.1. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$ -module. The relative weak assassin of $\mathcal{F}$ in $X$ over $S$ is the set

$\text{WeakAss}_{X/S}(\mathcal{F}) = \bigcup \nolimits _{s \in S} \text{WeakAss}(\mathcal{F}_ s)$

where $\mathcal{F}_ s = (X_ s \to X)^*\mathcal{F}$ is the restriction of $\mathcal{F}$ to the fibre of $f$ at $s$ .

Lemma 31.8.2. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$ -module. Then $\text{WeakAss}_{X/S}(\mathcal{F}) = \text{Ass}_{X/S}(\mathcal{F})$ .

Proof. This is true because the fibres of $f$ are locally Noetherian schemes, and associated and weakly associated points agree on locally Noetherian schemes, see Lemma 31.5.8. $\square$

Lemma 31.8.3. Let $f : X \to S$ be a morphism of schemes. Let $i : Z \to X$ be a finite morphism. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ Z$ -module. Then $\text{WeakAss}_{X/S}(i_*\mathcal{F}) = i(\text{WeakAss}_{Z/S}(\mathcal{F}))$ .

Proof. Let $i_ s : Z_ s \to X_ s$ be the induced morphism between fibres. Then $(i_*\mathcal{F})_ s = i_{s, *}(\mathcal{F}_ s)$ by Cohomology of Schemes, Lemma 30.5.1 and the fact that $i$ is affine. Hence we may apply Lemma 31.6.3 to conclude. $\square$

The Stacks project

31.8 Relative weak assassin

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