The Stacks project

75.6 A criterion for purity

This section is the analogue of More on Flatness, Section 38.18.

Lemma 75.6.1. Let $S$ be a scheme. Let $X$ be a decent algebraic space locally of finite type over $S$. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_ X$-module. Let $s \in S$ such that $\mathcal{F}$ is flat over $S$ at all points of $X_ s$. Let $x' \in \text{Ass}_{X/S}(\mathcal{F})$. If the closure of $\{ x'\} $ in $|X|$ meets $|X_ s|$, then the closure meets $\text{Ass}_{X/S}(\mathcal{F}) \cap |X_ s|$.

Proof. Observe that $|X_ s| \subset |X|$ is the set of points of $|X|$ lying over $s \in S$, see Decent Spaces, Lemma 66.18.6. Let $t \in |X_ s|$ be a specialization of $x'$ in $|X|$. Choose an affine scheme $U$ and a point $u \in U$ and an étale morphism $\varphi : U \to X$ mapping $u$ to $t$. By Decent Spaces, Lemma 66.12.2 we can choose a specialization $u' \leadsto u$ with $u'$ mapping to $x'$. Set $g = f \circ \varphi $. Observe that $s' = g(u') = f(x')$ specializes to $s$. By our definition of $\text{Ass}_{X/S}(\mathcal{F})$ we have $u' \in \text{Ass}_{U/S}(\varphi ^*\mathcal{F})$. By the schemes version of this lemma (More on Flatness, Lemma 38.18.1) we see that there is a specialization $u' \leadsto u$ with $u \in \text{Ass}_{U_ s}(\varphi ^*\mathcal{F}_ s) = \text{Ass}_{U/S}(\varphi ^*\mathcal{F}) \cap U_ s$. Hence $x = \varphi (u) \in \text{Ass}_{X/S}(\mathcal{F})$ lies over $s$ and the lemma is proved. $\square$

Lemma 75.6.2. Let $Y$ be an algebraic space over a scheme $S$. Let $g : X' \to X$ be a morphism of algebraic spaces over $Y$ with $X$ locally of finite type over $Y$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. If $\text{Ass}_{X/Y}(\mathcal{F}) \subset g(|X'|)$, then for any morphism $Z \to Y$ we have $\text{Ass}_{X_ Z/Z}(\mathcal{F}_ Z) \subset g_ Z(|X'_ Z|)$.

Proof. By Properties of Spaces, Lemma 64.4.3 the map $|X'_ Z| \to |X_ Z| \times _{|X|} |X'|$ is surjective as $X'_ Z$ is equal to $X_ Z \times _ X X'$. By Divisors on Spaces, Lemma 69.4.7 the map $|X_ Z| \to |X|$ sends $\text{Ass}_{X_ Z/Z}(\mathcal{F}_ Z)$ into $\text{Ass}_{X/Y}(\mathcal{F})$. The lemma follows. $\square$

Lemma 75.6.3. Let $Y$ be an algebraic space over a scheme $S$. Let $g : X' \to X$ be an étale morphism of algebraic spaces over $Y$. Assume the structure morphisms $X' \to Y$ and $X \to Y$ are decent and of finite type. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_ X$-module. Let $y \in |Y|$. Set $F = f^{-1}(\{ y\} ) \subset |X|$.

  1. If $\text{Ass}_{X/Y}(\mathcal{F}) \subset g(|X'|)$ and $g^*\mathcal{F}$ is (universally) pure above $y$, then $\mathcal{F}$ is (universally) pure above $y$.

  2. If $\mathcal{F}$ is pure above $y$, $g(|X'|)$ contains $F$, and $Y$ is affine local with closed point $y$, then $\text{Ass}_{X/Y}(\mathcal{F}) \subset g(|X'|)$.

  3. If $\mathcal{F}$ is pure above $y$, $\mathcal{F}$ is flat at all points of $F$, $g(|X'|)$ contains $\text{Ass}_{X/Y}(\mathcal{F}) \cap F$, and $Y$ is affine local with closed point $y$, then $\text{Ass}_{X/Y}(\mathcal{F}) \subset g(|X'|)$.

  4. Add more here.

Proof. The assumptions on $X \to Y$ and $X' \to Y$ guarantee that we may apply the material in Sections 75.2 and 75.3 to these morphisms and the sheaves $\mathcal{F}$ and $g^*\mathcal{F}$. Since $g$ is étale we see that $\text{Ass}_{X'/Y}(g^*\mathcal{F})$ is the inverse image of $\text{Ass}_{X/Y}(\mathcal{F})$ and the same remains true after base change.

Proof of (1). Assume $\text{Ass}_{X/Y}(\mathcal{F}) \subset g(|X'|)$. Suppose that $(T \to Y, t' \leadsto t, \xi )$ is an impurity of $\mathcal{F}$ above $y$. Since $\text{Ass}_{X_ T/T}(\mathcal{F}_ T) \subset g_ T(|X'_ T|)$ by Lemma 75.6.2 we can choose a point $\xi ' \in |X'_ T|$ mapping to $\xi $. By the above we see that $(T \to Y, t' \leadsto t, \xi ')$ is an impurity of $g^*\mathcal{F}$ above $y'$. This implies (1) is true.

Proof of (2). This follows from the fact that $g(|X'|)$ is open in $|X|$ and the fact that by purity every point of $\text{Ass}_{X/Y}(\mathcal{F})$ specializes to a point of $F$.

Proof of (3). This follows from the fact that $g(|X'|)$ is open in $|X|$ and the fact that by purity combined with Lemma 75.6.1 every point of $\text{Ass}_{X/Y}(\mathcal{F})$ specializes to a point of $\text{Ass}_{X/Y}(\mathcal{F}) \cap F$. $\square$

Lemma 75.6.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $y \in |Y|$. Assume

  1. $f$ is decent and of finite type,

  2. $\mathcal{F}$ is of finite type,

  3. $\mathcal{F}$ is flat over $Y$ at all points lying over $y$, and

  4. $\mathcal{F}$ is pure above $y$.

Then $\mathcal{F}$ is universally pure above $y$.

Proof. Consider the morphism $\mathop{\mathrm{Spec}}(\mathcal{O}_{Y, \overline{y}}) \to Y$. This is a flat morphism from the spectrum of a stricly henselian local ring which maps the closed point to $y$. By Lemma 75.3.4 we reduce to the case described in the next paragraph.

Assume $Y$ is the spectrum of a strictly henselian local ring $R$ with closed point $y$. By Lemma 75.4.6 there exists an étale morphism $g : X' \to X$ with $g(|X'|) \supset |X_ y|$, with $X'$ affine, and with $\Gamma (X', g^*\mathcal{F})$ a free $R$-module. Then $g^*\mathcal{F}$ is universally pure relative to $Y$, see More on Flatness, Lemma 38.17.4. Hence it suffices to prove that $g(|X'|)$ contains $\text{Ass}_{X/Y}(\mathcal{F})$ by Lemma 75.6.3 part (1). This in turn follows from Lemma 75.6.3 part (2). $\square$

Lemma 75.6.5. Let $S$ be a scheme. Let $f : X \to Y$ be a decent, finite type morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Assume $\mathcal{F}$ is flat over $Y$. In this case $\mathcal{F}$ is pure relative to $Y$ if and only if $\mathcal{F}$ is universally pure relative to $Y$.

Proof. Immediate consequence of Lemma 75.6.4 and the definitions. $\square$

Lemma 75.6.6. Let $Y$ be an algebraic space over a scheme $S$. Let $g : X' \to X$ be a flat morphism of algebraic spaces over $Y$ with $X$ locally of finite type over $Y$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module which is flat over $Y$. If $\text{Ass}_{X/Y}(\mathcal{F}) \subset g(|X'|)$ then the canonical map

\[ \mathcal{F} \longrightarrow g_*g^*\mathcal{F} \]

is injective, and remains injective after any base change.

Proof. The final assertion means that $\mathcal{F}_ Z \to (g_ Z)_*g_ Z^*\mathcal{F}_ Z$ is injective for any morphism $Z \to Y$. Since the assumption on the relative assassin is preserved by base change (Lemma 75.6.2) it suffices to prove the injectivity of the displayed arrow.

Let $\mathcal{K} = \mathop{\mathrm{Ker}}(\mathcal{F} \to g_*g^*\mathcal{F})$. Our goal is to prove that $\mathcal{K} = 0$. In order to do this it suffices to prove that $\text{WeakAss}_ X(\mathcal{K}) = \emptyset $, see Divisors on Spaces, Lemma 69.2.5. We have $\text{WeakAss}_ X(\mathcal{K}) \subset \text{WeakAss}_ X(\mathcal{F})$, see Divisors on Spaces, Lemma 69.2.4. As $\mathcal{F}$ is flat we see from Lemma 75.4.4 that $\text{WeakAss}_ X(\mathcal{F}) \subset \text{Ass}_{X/Y}(\mathcal{F})$. By assumption any point $x$ of $\text{Ass}_{X/Y}(\mathcal{F})$ is the image of some $x' \in |X'|$. Since $g$ is flat the local ring map $\mathcal{O}_{X, \overline{x}} \to \mathcal{O}_{X', \overline{x}'}$ is faithfully flat, hence the map

\[ \mathcal{F}_{\overline{x}} \longrightarrow (g^*\mathcal{F})_{\overline{x}'} = \mathcal{F}_{\overline{x}} \otimes _{\mathcal{O}_{X, \overline{x}}} \mathcal{O}_{X', \overline{x}'} \]

is injective (see Algebra, Lemma 10.81.11). Since the displayed arrow factors through $\mathcal{F}_{\overline{x}} \to (g_*g^*\mathcal{F})_{\overline{x}}$, we conclude that $\mathcal{K}_{\overline{x}} = 0$. Hence $x$ cannot be a weakly associated point of $\mathcal{K}$ and we win. $\square$


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