77.3 Relatively pure modules
This section is the analogue of More on Flatness, Section 38.16.
Definition 77.3.1. In Situation 77.2.1.
We say \mathcal{F} is pure above y if none of the equivalent conditions of Lemma 77.2.5 hold.
We say \mathcal{F} is universally pure above y if there does not exist any impurity of \mathcal{F} above y.
We say that X is pure above y if \mathcal{O}_ X is pure above y.
We say \mathcal{F} is universally Y-pure, or universally pure relative to Y if \mathcal{F} is universally pure above y for every y \in |Y|.
We say \mathcal{F} is Y-pure, or pure relative to Y if \mathcal{F} is pure above y for every y \in |Y|.
We say that X is Y-pure or pure relative to Y if \mathcal{O}_ X is pure relative to Y.
The obligatory lemmas follow.
Lemma 77.3.2. In Situation 77.2.1.
\mathcal{F} is universally pure above y, and
for every morphism (Y', y') \to (Y, y) of pointed algebraic spaces the pullback \mathcal{F}_{Y'} is pure above y'.
In particular, \mathcal{F} is universally pure relative to Y if and only if every base change \mathcal{F}_{Y'} of \mathcal{F} is pure relative to Y'.
Proof.
This is formal.
\square
Lemma 77.3.3. In Situation 77.2.1. Let (Y', y') \to (Y, y) be a morphism of pointed algebraic spaces. If Y' \to Y is quasi-finite at y' and \mathcal{F} is pure above y, then \mathcal{F}_{Y'} is pure above y'.
Proof.
It (T \to Y', t' \leadsto t, \xi ) is an impurity of \mathcal{F}_{Y'} above y' with T \to Y' quasi-finite at t, then (T \to Y, t' \to t, \xi ) is an impurity of \mathcal{F} above y with T \to Y quasi-finite at t, see Morphisms of Spaces, Lemma 67.27.3. Hence the lemma follows immediately from the definition of purity.
\square
Purity satisfies flat descent.
Lemma 77.3.4. In Situation 77.2.1. Let (Y_1, y_1) \to (Y, y) be a morphism of pointed algebraic spaces. Assume Y_1 \to Y is flat at y_1.
If \mathcal{F}_{Y_1} is pure above y_1, then \mathcal{F} is pure above y.
If \mathcal{F}_{Y_1} is universally pure above y_1, then \mathcal{F} is universally pure above y.
Proof.
This is true because impurities go up along a flat base change, see Lemma 77.2.4. For example part (1) follows because by any impurity (T \to Y, t' \leadsto t, \xi ) of \mathcal{F} above y with T \to Y quasi-finite at t by the lemma leads to an impurity (T_1 \to Y_1, t_1' \leadsto t_1, \xi _1) of the pullback \mathcal{F}_1 of \mathcal{F} to X_1 = Y_1 \times _ Y X over y_1 such that T_1 is étale over Y_1 \times _ Y T. Hence T_1 \to Y_1 is quasi-finite at t_1 because étale morphisms are locally quasi-finite and compositions of locally quasi-finite morphisms are locally quasi-finite (Morphisms of Spaces, Lemmas 67.39.5 and 67.27.3). Similarly for part (2).
\square
Lemma 77.3.5. In Situation 77.2.1. Let i : Z \to X be a closed immersion and assume that \mathcal{F} = i_*\mathcal{G} for some finite type, quasi-coherent sheaf \mathcal{G} on Z. Then \mathcal{G} is (universally) pure above y if and only if \mathcal{F} is (universally) pure above y.
Proof.
This follows from Divisors on Spaces, Lemma 71.4.9.
\square
Lemma 77.3.6. In Situation 77.2.1.
If the support of \mathcal{F} is proper over Y, then \mathcal{F} is universally pure relative to Y.
If f is proper, then \mathcal{F} is universally pure relative to Y.
If f is proper, then X is universally pure relative to Y.
Proof.
First we reduce (1) to (2). Namely, let Z \subset X be the scheme theoretic support of \mathcal{F} (Morphisms of Spaces, Definition 67.15.4). Let i : Z \to X be the corresponding closed immersion and write \mathcal{F} = i_*\mathcal{G} for some finite type quasi-coherent \mathcal{O}_ Z-module \mathcal{G}. In case (1) Z \to Y is proper by assumption. Thus by Lemma 77.3.5 case (1) reduces to case (2).
Assume f is proper. Let (g : T \to Y, t' \leadsto t, \xi ) be an impurity of \mathcal{F} above y. Since f is proper, it is universally closed. Hence f_ T : X_ T \to T is closed. Since f_ T(\xi ) = t' this implies that t \in f(\overline{\{ \xi \} }) which is a contradiction.
\square
Comments (0)