The Stacks project

Lemma 76.4.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $x \in |X|$ with image $y \in |Y|$. Let $\mathcal{F}$ be a finite type quasi-coherent sheaf on $X$. Let $\mathcal{G}$ be a quasi-coherent sheaf on $Y$. If $\mathcal{F}$ is flat at $x$ over $Y$, then

\[ x \in \text{WeakAss}_ X(\mathcal{F} \otimes _{\mathcal{O}_ X} f^*\mathcal{G}) \Leftrightarrow y \in \text{WeakAss}_ Y(\mathcal{G}) \text{ and } x \in \text{Ass}_{X/Y}(\mathcal{F}). \]

Proof. Choose a commutative diagram

\[ \xymatrix{ U \ar[d] \ar[r]_ g & V \ar[d] \\ X \ar[r]^ f & Y } \]

where $U$ and $V$ are schemes and the vertical arrows are surjective ├ętale. Choose $u \in U$ mapping to $x$. Let $\mathcal{E} = \mathcal{F}|_ U$ and $\mathcal{H} = \mathcal{G}|_ V$. Let $v \in V$ be the image of $u$. Then $x \in \text{WeakAss}_ X(\mathcal{F} \otimes _{\mathcal{O}_ X} f^*\mathcal{G})$ if and only if $u \in \text{WeakAss}_ X(\mathcal{E} \otimes _{\mathcal{O}_ X} g^*\mathcal{H})$ by Divisors on Spaces, Definition 70.2.2. Similarly, $y \in \text{WeakAss}_ Y(\mathcal{G})$ if and only if $v \in \text{WeakAss}_ V(\mathcal{H})$. Finally, we have $x \in \text{Ass}_{X/Y}(\mathcal{F})$ if and only if $u \in \text{Ass}_{U_ v}(\mathcal{E}|_{U_ v})$ by Divisors on Spaces, Definition 70.4.5. Observe that flatness of $\mathcal{F}$ at $x$ is equivalent to flatness of $\mathcal{E}$ at $u$, see Morphisms of Spaces, Definition 66.31.2. The equivalence for $g : U \to V$, $\mathcal{E}$, $\mathcal{H}$, $u$, and $v$ is More on Flatness, Lemma 38.13.3. $\square$


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