70.2 Associated and weakly associated points

In the case of schemes we have introduced two competing notions of associated points. Namely, the usual associated points (Divisors, Section 31.2) and the weakly associated points (Divisors, Section 31.5). For a general algebraic space the notion of an associated point is basically useless and we don't even bother to introduce it. If the algebraic space is locally Noetherian, then we allow ourselves to use the phrase “associated point” instead of “weakly associated point” as the notions are the same for Noetherian schemes (Divisors, Lemma 31.5.8). Before we make our definition, we need a lemma.

Lemma 70.2.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $x \in |X|$. The following are equivalent

1. for some étale morphism $f : U \to X$ with $U$ a scheme and $u \in U$ mapping to $x$, the point $u$ is weakly associated to $f^*\mathcal{F}$,

2. for every étale morphism $f : U \to X$ with $U$ a scheme and $u \in U$ mapping to $x$, the point $u$ is weakly associated to $f^*\mathcal{F}$,

3. the maximal ideal of $\mathcal{O}_{X, \overline{x}}$ is a weakly associated prime of the stalk $\mathcal{F}_{\overline{x}}$.

If $X$ is locally Noetherian, then these are also equivalent to

1. for some étale morphism $f : U \to X$ with $U$ a scheme and $u \in U$ mapping to $x$, the point $u$ is associated to $f^*\mathcal{F}$,

2. for every étale morphism $f : U \to X$ with $U$ a scheme and $u \in U$ mapping to $x$, the point $u$ is associated to $f^*\mathcal{F}$,

3. the maximal ideal of $\mathcal{O}_{X, \overline{x}}$ is an associated prime of the stalk $\mathcal{F}_{\overline{x}}$.

Proof. Choose a scheme $U$ with a point $u$ and an étale morphism $f : U \to X$ mapping $u$ to $x$. Lift $\overline{x}$ to a geometric point of $U$ over $u$. Recall that $\mathcal{O}_{X, \overline{x}} = \mathcal{O}_{U, u}^{sh}$ where the strict henselization is with respect to our chosen lift of $\overline{x}$, see Properties of Spaces, Lemma 65.22.1. Finally, we have

$\mathcal{F}_{\overline{x}} = (f^*\mathcal{F})_ u \otimes _{\mathcal{O}_{U, u}} \mathcal{O}_{X, \overline{x}} = (f^*\mathcal{F})_ u \otimes _{\mathcal{O}_{U, u}} \mathcal{O}_{U, u}^{sh}$

by Properties of Spaces, Lemma 65.29.4. Hence the equivalence of (1), (2), and (3) follows from More on Flatness, Lemma 38.2.9. If $X$ is locally Noetherian, then any $U$ as above is locally Noetherian, hence we see that (1), resp. (2) are equivalent to (4), resp. (5) by Divisors, Lemma 31.5.8. On the other hand, in the locally Noetherian case the local ring $\mathcal{O}_{X, \overline{x}}$ is Noetherian too (Properties of Spaces, Lemma 65.24.4). Hence the equivalence of (3) and (6) by the same lemma (or by Algebra, Lemma 10.66.9). $\square$

Definition 70.2.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $x \in |X|$.

1. We say $x$ is weakly associated to $\mathcal{F}$ if the equivalent conditions (1), (2), and (3) of Lemma 70.2.1 are satisfied.

2. We denote $\text{WeakAss}(\mathcal{F})$ the set of weakly associated points of $\mathcal{F}$.

3. The weakly associated points of $X$ are the weakly associated points of $\mathcal{O}_ X$.

If $X$ is locally Noetherian we will say $x$ is associated to $\mathcal{F}$ if and only if $x$ is weakly associated to $\mathcal{F}$ and we set $\text{Ass}(\mathcal{F}) = \text{WeakAss}(\mathcal{F})$. Finally (still assuming $X$ is locally Noetherian), we will say $x$ is an associated point of $X$ if and only if $x$ is a weakly associated point of $X$.

At this point we can prove the obligatory lemmas.

Lemma 70.2.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Then $\text{WeakAss}(\mathcal{F}) \subset \text{Supp}(\mathcal{F})$.

Proof. This is immediate from the definitions. The support of an abelian sheaf on $X$ is defined in Properties of Spaces, Definition 65.20.3. $\square$

Lemma 70.2.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ be a short exact sequence of quasi-coherent sheaves on $X$. Then $\text{WeakAss}(\mathcal{F}_2) \subset \text{WeakAss}(\mathcal{F}_1) \cup \text{WeakAss}(\mathcal{F}_3)$ and $\text{WeakAss}(\mathcal{F}_1) \subset \text{WeakAss}(\mathcal{F}_2)$.

Proof. For every geometric point $\overline{x} \in X$ the sequence of stalks $0 \to \mathcal{F}_{1, \overline{x}} \to \mathcal{F}_{2, \overline{x}} \to \mathcal{F}_{3, \overline{x}} \to 0$ is a short exact sequence of $\mathcal{O}_{X, \overline{x}}$-modules. Hence the lemma follows from Algebra, Lemma 10.66.4. $\square$

Lemma 70.2.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Then

$\mathcal{F} = (0) \Leftrightarrow \text{WeakAss}(\mathcal{F}) = \emptyset$

Proof. Choose a scheme $U$ and a surjective étale morphism $f : U \to X$. Then $\mathcal{F}$ is zero if and only if $f^*\mathcal{F}$ is zero. Hence the lemma follows from the definition and the lemma in the case of schemes, see Divisors, Lemma 31.5.5. $\square$

Lemma 70.2.6. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $x \in |X|$. If

1. $x \in \text{Supp}(\mathcal{F})$

2. $x$ is a codimension $0$ point of $X$ (Properties of Spaces, Definition 65.10.2).

Then $x \in \text{WeakAss}(\mathcal{F})$. If $\mathcal{F}$ is a finite type $\mathcal{O}_ X$-module with scheme theoretic support $Z$ (Morphisms of Spaces, Definition 66.15.4) and $x$ is a codimension $0$ point of $Z$, then $x \in \text{WeakAss}(\mathcal{F})$.

Proof. Since $x \in \text{Supp}(\mathcal{F})$ the stalk $\mathcal{F}_{\overline{x}}$ is not zero. Hence $\text{WeakAss}(\mathcal{F}_{\overline{x}})$ is nonempty by Algebra, Lemma 10.66.5. On the other hand, the spectrum of $\mathcal{O}_{X, \overline{x}}$ is a singleton. Hence $x$ is a weakly associated point of $\mathcal{F}$ by definition. The final statement follows as $\mathcal{O}_{X, \overline{x}} \to \mathcal{O}_{Z, \overline{z}}$ is a surjection, the spectrum of $\mathcal{O}_{Z, \overline{z}}$ is a singleton, and $\mathcal{F}_{\overline{x}}$ is a nonzero module over $\mathcal{O}_{Z, \overline{z}}$. $\square$

Lemma 70.2.7. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $x \in |X|$. If

1. $X$ is decent (for example quasi-separated or locally separated),

2. $x \in \text{Supp}(\mathcal{F})$

3. $x$ is not a specialization of another point in $\text{Supp}(\mathcal{F})$.

Then $x \in \text{WeakAss}(\mathcal{F})$.

Proof. (A quasi-separated algebraic space is decent, see Decent Spaces, Section 67.6. A locally separated algebraic space is decent, see Decent Spaces, Lemma 67.15.2.) Choose a scheme $U$, a point $u \in U$, and an étale morphism $f : U \to X$ mapping $u$ to $x$. By Decent Spaces, Lemma 67.12.1 if $u' \leadsto u$ is a nontrivial specialization, then $f(u') \not= x$. Hence we see that $u \in \text{Supp}(f^*\mathcal{F})$ is not a specialization of another point of $\text{Supp}(f^*\mathcal{F})$. Hence $u \in \text{WeakAss}(f^*\mathcal{F})$ by Divisors, Lemma 70.2.6. $\square$

Lemma 70.2.8. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Then $\text{Ass}(\mathcal{F}) \cap W$ is finite for every quasi-compact open $W \subset |X|$.

Proof. Choose a quasi-compact scheme $U$ and an étale morphism $U \to X$ such that $W$ is the image of $|U| \to |X|$. Then $U$ is a Noetherian scheme and we may apply Divisors, Lemma 31.2.5 to conclude. $\square$

Lemma 70.2.9. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. If $U \to X$ is an étale morphism such that $\text{WeakAss}(\mathcal{F}) \subset \mathop{\mathrm{Im}}(|U| \to |X|)$, then $\Gamma (X, \mathcal{F}) \to \Gamma (U, \mathcal{F})$ is injective.

Proof. Let $s \in \Gamma (X, \mathcal{F})$ be a section which restricts to zero on $U$. Let $\mathcal{F}' \subset \mathcal{F}$ be the image of the map $\mathcal{O}_ X \to \mathcal{F}$ defined by $s$. Then $\mathcal{F}'|_ U = 0$. This implies that $\text{WeakAss}(\mathcal{F}') \cap \mathop{\mathrm{Im}}(|U| \to |X|) = \emptyset$ (by the definition of weakly associated points). On the other hand, $\text{WeakAss}(\mathcal{F}') \subset \text{WeakAss}(\mathcal{F})$ by Lemma 70.2.4. We conclude $\text{WeakAss}(\mathcal{F}') = \emptyset$. Hence $\mathcal{F}' = 0$ by Lemma 70.2.5. $\square$

Lemma 70.2.10. Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $y \in |Y|$ be a point which is not in the image of $|f|$. Then $y$ is not weakly associated to $f_*\mathcal{F}$.

Proof. By Morphisms of Spaces, Lemma 66.11.2 the $\mathcal{O}_ Y$-module $f_*\mathcal{F}$ is quasi-coherent hence the lemma makes sense. Choose an affine scheme $V$, a point $v \in V$, and an étale morphism $V \to Y$ mapping $v$ to $y$. We may replace $f : X \to Y$, $\mathcal{F}$, $y$ by $X \times _ Y V \to V$, $\mathcal{F}|_{X \times _ Y V}$, $v$. Thus we may assume $Y$ is an affine scheme. In this case $X$ is quasi-compact, hence we can choose an affine scheme $U$ and a surjective étale morphism $U \to X$. Denote $g : U \to Y$ the composition. Then $f_*\mathcal{F} \subset g_*(\mathcal{F}|_ U)$. By Lemma 70.2.4 we reduce to the case of schemes which is Divisors, Lemma 31.5.9. $\square$

Lemma 70.2.11. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of quasi-coherent $\mathcal{O}_ X$-modules. Assume that for every $x \in |X|$ at least one of the following happens

1. $\mathcal{F}_{\overline{x}} \to \mathcal{G}_{\overline{x}}$ is injective, or

2. $x \not\in \text{WeakAss}(\mathcal{F})$.

Then $\varphi$ is injective.

Proof. The assumptions imply that $\text{WeakAss}(\mathop{\mathrm{Ker}}(\varphi )) = \emptyset$ and hence $\mathop{\mathrm{Ker}}(\varphi ) = 0$ by Lemma 70.2.5. $\square$

Lemma 70.2.12. Let $S$ be a scheme. Let $X$ be a reduced algebraic space over $S$. Then the weakly associated point of $X$ are exactly the codimension $0$ points of $X$.

Proof. Working étale locally this follows from Divisors, Lemma 31.5.12 and Properties of Spaces, Lemma 65.11.1. $\square$

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