Lemma 77.8.5. In Situation 77.7.1. Assume
$f$ is of finite presentation, and
$\mathcal{G}$ is of finite presentation, flat over $B$, and pure relative to $B$.
Then $F_{zero}$ is an algebraic space and $F_{zero} \to B$ is a closed immersion. If $\mathcal{F}$ is of finite type, then $F_{zero} \to B$ is of finite presentation.
Proof. By Lemma 77.6.5 the module $\mathcal{G}$ is universally pure relative to $B$. In order to prove that $F_{zero}$ is an algebraic space, it suffices to show that $F_{zero} \to B$ is representable, see Spaces, Lemma 65.11.3. Let $B' \to B$ be a morphism where $B'$ is a scheme and let $u' : \mathcal{F}' \to \mathcal{G}'$ be the pullback of $u$ to $X' = X_{B'}$. Then the associated functor $F'_{zero}$ equals $F_{zero} \times _ B B'$. This reduces us to the case that $B$ is a scheme.
Assume $B$ is a scheme. We will show that $F_{zero}$ is representable by a closed subscheme of $B$. By Lemma 77.7.2 and Descent, Lemmas 35.37.2 and 35.39.1 the question is local for the étale topology on $B$. Let $b \in B$. We first replace $B$ by an affine neighbourhood of $b$. Choose a diagram
and $b' \in B'$ mapping to $b \in B$ as in Lemma 77.5.2. As we are working étale locally, we may replace $B$ by $B'$ and assume that we have a diagram
with $B$ and $X'$ affine such that $\Gamma (X', g^*\mathcal{G})$ is a projective $\Gamma (B, \mathcal{O}_ B)$-module and $g(|X'|) \supset |X_ b|$. Let $U \subset X$ be the open subspace with $|U| = g(|X'|)$. By Divisors on Spaces, Lemma 71.4.10 the set
is constructible in $B$. By Lemma 77.6.3 part (2) we see that $E$ contains $\mathop{\mathrm{Spec}}(\mathcal{O}_{B, b})$. By Morphisms, Lemma 29.22.4 we see that $E$ contains an open neighbourhood of $b$. Hence after replacing $B$ by a smaller affine neighbourhood of $b$ we may assume that $\text{Ass}_{X/B}(\mathcal{G}) \subset g(|X'|)$.
From Lemma 77.6.6 it follows that $u : \mathcal{F} \to \mathcal{G}$ is injective if and only if $g^*u : g^*\mathcal{F} \to g^*\mathcal{G}$ is injective, and the same remains true after any base change. Hence we have reduced to the case where, in addition to the assumptions in the theorem, $X \to B$ is a morphism of affine schemes and $\Gamma (X, \mathcal{G})$ is a projective $\Gamma (B, \mathcal{O}_ B)$-module. This case follows immediately from Lemma 77.8.3.
We still have to show that $F_{zero} \to B$ is of finite presentation if $\mathcal{F}$ is of finite type. This follows from Lemma 77.7.4 combined with Limits of Spaces, Proposition 70.3.10. $\square$
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)