The Stacks project

Lemma 37.18.4. 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 of finite type. Let $y \in \text{Ass}_{X/S}(\mathcal{F})$. Denote $Y \subset X$ the closure of $\{ y\} $ in $X$ viewed as an integral closed subscheme. Denote $T \subset S$ the closure of $\{ f(y)\} $ viewed as an integral closed subscheme. We obtain a commutative diagram

\[ \xymatrix{ Y \ar[r] \ar[d] & X \ar[d] \\ T \ar[r] & S } \]

where $Y \to T$ is dominant. Assume $\mathcal{F}$ is flat over $S$ at all generic points of irreducible components of fibres of $Y \to T$ (for example if $\mathcal{F}$ is flat over $S$). Then

  1. if $s \in S$ and $x \in Y_ s$ is the generic point of an irreducible component of $Y_ s$, then $x \in \text{Ass}_{X/S}(\mathcal{F})$, and

  2. there is an integer $d \geq 0$ such that $Y \to T$ is of relative dimension $d$, see Morphisms, Definition 29.29.1.

Proof. This follows immediately from the pointwise version Lemma 37.18.3. Note that to compute the dimension of the locally algebraic schemes $Y_ s$ it suffices to look near the generic points, see Varieties, Section 33.20. $\square$

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