The Stacks project

Lemma 76.7.4. In Situation 76.7.1.

  1. If $\mathcal{G}$ is of finite type and the scheme theoretic support of $\mathcal{G}$ is quasi-compact over $B$, then $F_{surj}$ is limit preserving.

  2. If $\mathcal{F}$ of finite type and the scheme theoretic support of $\mathcal{F}$ is quasi-compact over $B$, then $F_{zero}$ is limit preserving.

  3. If $\mathcal{F}$ is of finite type, $\mathcal{G}$ is of finite presentation, and the scheme theoretic supports of $\mathcal{F}$ and $\mathcal{G}$ are quasi-compact over $B$, then $F_{iso}$ is limit preserving.

Proof. Proof of (1). Let $i : Z \to X$ be the scheme theoretic support of $\mathcal{G}$ and think of $\mathcal{G}$ as a finite type quasi-coherent module on $Z$. We may replace $X$ by $Z$ and $u$ by the map $i^*\mathcal{F} \to \mathcal{G}$ (details omitted). Hence we may assume $f$ is quasi-compact and $\mathcal{G}$ of finite type. Let $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ be a directed limit of affine $B$-schemes and assume that $u_ T$ is surjective. Set $X_ i = X_{T_ i} = X \times _ S T_ i$ and $u_ i = u_{T_ i} : \mathcal{F}_ i = \mathcal{F}_{T_ i} \to \mathcal{G}_ i = \mathcal{G}_{T_ i}$. To prove (1) we have to show that $u_ i$ is surjective for some $i$. Pick $0 \in I$ and replace $I$ by $\{ i \mid i \geq 0\} $. Since $f$ is quasi-compact we see $X_0$ is quasi-compact. Hence we may choose a surjective étale morphism $\varphi _0 : W_0 \to X_0$ where $W_0$ is an affine scheme. Set $W = W_0 \times _{T_0} T$ and $W_ i = W_0 \times _{T_0} T_ i$ for $i \geq 0$. These are affine schemes endowed with a surjective étale morphisms $\varphi : W \to X_ T$ and $\varphi _ i : W_ i \to X_ i$. Note that $W = \mathop{\mathrm{lim}}\nolimits W_ i$. Hence $\varphi ^*u_ T$ is surjective and it suffices to prove that $\varphi _ i^*u_ i$ is surjective for some $i$. Thus we have reduced the problem to the affine case which is Algebra, Lemma 10.127.5 part (2).

Proof of (2). Assume $\mathcal{F}$ is of finite type with scheme theoretic support $Z \subset B$ quasi-compact over $B$. Let $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ be a directed limit of affine $B$-schemes and assume that $u_ T$ is zero. Set $X_ i = T_ i \times _ B X$ and denote $u_ i : \mathcal{F}_ i \to \mathcal{G}_ i$ the pullback. Choose $0 \in I$ and replace $I$ by $\{ i \mid i \geq 0\} $. Set $Z_0 = Z \times _ X X_0$. By Morphisms of Spaces, Lemma 66.15.2 the support of $\mathcal{F}_ i$ is $|Z_0|$. Since $|Z_0|$ is quasi-compact we can find an affine scheme $W_0$ and an étale morphism $W_0 \to X_0$ such that $|Z_0| \subset \mathop{\mathrm{Im}}(|W_0| \to |X_0|)$. Set $W = W_0 \times _{T_0} T$ and $W_ i = W_0 \times _{T_0} T_ i$ for $i \geq 0$. These are affine schemes endowed with étale morphisms $\varphi : W \to X_ T$ and $\varphi _ i : W_ i \to X_ i$. Note that $W = \mathop{\mathrm{lim}}\nolimits W_ i$ and that the support of $\mathcal{F}_ T$ and $\mathcal{F}_ i$ is contained in the image of $|W| \to |X_ T|$ and $|W_ i| \to |X_ i|$. Now $\varphi ^*u_ T$ is injective and it suffices to prove that $\varphi _ i^*u_ i$ is injective for some $i$. Thus we have reduced the problem to the affine case which is Algebra, Lemma 10.127.5 part (1).

Proof of (3). This can be proven in exactly the same manner as in the previous two paragraphs using Algebra, Lemma 10.127.5 part (3). We can also deduce it from (1) and (2) as follows. Let $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ be a directed limit of affine $B$-schemes and assume that $u_ T$ is an isomorphism. By part (1) there exists an $0 \in I$ such that $u_{T_0}$ is surjective. Set $\mathcal{K} = \mathop{\mathrm{Ker}}(u_{T_0})$ and consider the map of quasi-coherent modules $v : \mathcal{K} \to \mathcal{F}_{T_0}$. For $i \geq 0$ the base change $v_{T_ i}$ is zero if and only if $u_ i$ is an isomorphism. Moreover, $v_ T$ is zero. Since $\mathcal{G}_{T_0}$ is of finite presentation, $\mathcal{F}_{T_0}$ is of finite type, and $u_{T_0}$ is surjective we conclude that $\mathcal{K}$ is of finite type (Modules on Sites, Lemma 18.24.1). It is clear that the support of $\mathcal{K}$ is contained in the support of $\mathcal{F}_{T_0}$ which is quasi-compact over $T_0$. Hence we can apply part (2) to see that $v_{T_ i}$ is zero for some $i$. $\square$


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