## 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.
$\square$

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