The Stacks project

Lemma 76.8.5. In Situation 76.7.1. Assume

  1. $f$ is of finite presentation, and

  2. $\mathcal{G}$ 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 76.6.5 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 64.11.3. 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 76.7.2 and Descent, Lemmas 35.37.2 and 35.39.1 the question is local for the ├ętale topology on $B$. Let $b \in B$. We first replace $B$ by an affine neighbourhood of $b$. Choose a diagram

\[ \xymatrix{ X \ar[d] & X' \ar[l]^ g \ar[d] \\ B & B' \ar[l] } \]

and $b' \in B'$ mapping to $b \in B$ as in Lemma 76.5.2. As we are working ├ętale locally, we may replace $B$ by $B'$ and assume that we have a diagram

\[ \xymatrix{ X \ar[rd] & & X' \ar[ll]^ g \ar[ld] \\ & B } \]

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 70.4.10 the set

\[ E = \{ t \in B : \text{Ass}_{X_ t}(\mathcal{G}_ t) \subset |U_ t|\} = \{ t \in B : \text{Ass}_{X/B}(\mathcal{G}) \cap |X_ t| \subset |U_ t|\} \]

is constructible in $B$. By Lemma 76.6.3 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 76.6.6 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 76.8.3.

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 76.7.4 combined with Limits of Spaces, Proposition 69.3.10. $\square$

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