Lemma 71.4.10. Let $Y$ be a scheme. Let $X$ be an algebraic space of finite presentation over $Y$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite presentation. Let $U \subset X$ be an open subspace such that $U \to Y$ is quasi-compact. Then the set
is locally constructible in $Y$.
Proof. Note that since $Y$ is a scheme, it makes sense to take the fibres $X_ y = \mathop{\mathrm{Spec}}(\kappa (y)) \times _ Y X$. (Also, by our definitions, the set $\text{Ass}_{X_ y}(\mathcal{F}_ y)$ is exactly the fibre of $\text{Ass}_{X/Y}(\mathcal{F}) \to Y$ over $y$, but we won't need this.) The question is local on $Y$, indeed, we have to show that $E$ is constructible if $Y$ is affine. In this case $X$ is quasi-compact. Choose an affine scheme $W$ and a surjective étale morphism $\varphi : W \to X$. Then $\text{Ass}_{X_ y}(\mathcal{F}_ y)$ is the image of $\text{Ass}_{W_ y}(\varphi ^*\mathcal{F}_ y)$ for all $y \in Y$. Hence the lemma follows from the case of schemes for the open $\varphi ^{-1}(U) \subset W$ and the morphism $W \to Y$. The case of schemes is More on Morphisms, Lemma 37.25.5. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)