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30.2. Associated points

Let $R$ be a ring and let $M$ be an $R$-module. Recall that a prime $\mathfrak p \subset R$ is associated to $M$ if there exists an element of $M$ whose annihilator is $\mathfrak p$. See Algebra, Definition 10.62.1. Here is the definition of associated points for quasi-coherent sheaves on schemes as given in [EGA, IV Definition 3.1.1].

Definition 30.2.1. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.

  1. We say $x \in X$ is associated to $\mathcal{F}$ if the maximal ideal $\mathfrak m_x$ is associated to the $\mathcal{O}_{X, x}$-module $\mathcal{F}_x$.
  2. We denote $\text{Ass}(\mathcal{F})$ or $\text{Ass}_X(\mathcal{F})$ the set of associated points of $\mathcal{F}$.
  3. The associated points of $X$ are the associated points of $\mathcal{O}_X$.

These definitions are most useful when $X$ is locally Noetherian and $\mathcal{F}$ of finite type. For example it may happen that a generic point of an irreducible component of $X$ is not associated to $X$, see Example 30.2.7. In the non-Noetherian case it may be more convenient to use weakly associated points, see Section 30.5. Let us link the scheme theoretic notion with the algebraic notion on affine opens; note that this correspondence works perfectly only for locally Noetherian schemes.

Lemma 30.2.2. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $\mathop{\rm Spec}(A) = U \subset X$ be an affine open, and set $M = \Gamma(U, \mathcal{F})$. Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime.

  1. If $\mathfrak p$ is associated to $M$, then $x$ is associated to $\mathcal{F}$.
  2. If $\mathfrak p$ is finitely generated, then the converse holds as well.

In particular, if $X$ is locally Noetherian, then the equivalence $$ \mathfrak p \in \text{Ass}(M) \Leftrightarrow x \in \text{Ass}(\mathcal{F}) $$ holds for all pairs $(\mathfrak p, x)$ as above.

Proof. This follows from Algebra, Lemma 10.62.15. But we can also argue directly as follows. Suppose $\mathfrak p$ is associated to $M$. Then there exists an $m \in M$ whose annihilator is $\mathfrak p$. Since localization is exact we see that $\mathfrak pA_{\mathfrak p}$ is the annihilator of $m/1 \in M_{\mathfrak p}$. Since $M_{\mathfrak p} = \mathcal{F}_x$ (Schemes, Lemma 25.5.4) we conclude that $x$ is associated to $\mathcal{F}$.

Conversely, assume that $x$ is associated to $\mathcal{F}$, and $\mathfrak p$ is finitely generated. As $x$ is associated to $\mathcal{F}$ there exists an element $m' \in M_{\mathfrak p}$ whose annihilator is $\mathfrak pA_{\mathfrak p}$. Write $m' = m/f$ for some $f \in A$, $f \not \in \mathfrak p$. The annihilator $I$ of $m$ is an ideal of $A$ such that $IA_{\mathfrak p} = \mathfrak pA_{\mathfrak p}$. Hence $I \subset \mathfrak p$, and $(\mathfrak p/I)_{\mathfrak p} = 0$. Since $\mathfrak p$ is finitely generated, there exists a $g \in A$, $g \not \in \mathfrak p$ such that $g(\mathfrak p/I) = 0$. Hence the annihilator of $gm$ is $\mathfrak p$ and we win.

If $X$ is locally Noetherian, then $A$ is Noetherian (Properties, Lemma 27.5.2) and $\mathfrak p$ is always finitely generated. $\square$

Lemma 30.2.3. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then $\text{Ass}(\mathcal{F}) \subset \text{Supp}(\mathcal{F})$.

Proof. This is immediate from the definitions. $\square$

Lemma 30.2.4. Let $X$ be a scheme. 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 $\text{Ass}(\mathcal{F}_2) \subset \text{Ass}(\mathcal{F}_1) \cup \text{Ass}(\mathcal{F}_3)$ and $\text{Ass}(\mathcal{F}_1) \subset \text{Ass}(\mathcal{F}_2)$.

Proof. For every point $x \in X$ the sequence of stalks $0 \to \mathcal{F}_{1, x} \to \mathcal{F}_{2, x} \to \mathcal{F}_{3, x} \to 0$ is a short exact sequence of $\mathcal{O}_{X, x}$-modules. Hence the lemma follows from Algebra, Lemma 10.62.3. $\square$

Lemma 30.2.5. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Then $\text{Ass}(\mathcal{F}) \cap U$ is finite for every quasi-compact open $U \subset X$.

Proof. This is true because the set of associated primes of a finite module over a Noetherian ring is finite, see Algebra, Lemma 10.62.5. To translate from schemes to algebra use that $U$ is a finite union of affine opens, each of these opens is the spectrum of a Noetherian ring (Properties, Lemma 27.5.2), $\mathcal{F}$ corresponds to a finite module over this ring (Cohomology of Schemes, Lemma 29.9.1), and finally use Lemma 30.2.2. $\square$

Lemma 30.2.6. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then $$ \mathcal{F} = 0 \Leftrightarrow \text{Ass}(\mathcal{F}) = \emptyset. $$

Proof. If $\mathcal{F} = 0$, then $\text{Ass}(\mathcal{F}) = \emptyset$ by definition. Conversely, if $\text{Ass}(\mathcal{F}) = \emptyset$, then $\mathcal{F} = 0$ by Algebra, Lemma 10.62.7. To translate from schemes to algebra, restrict to any affine and use Lemma 30.2.2. $\square$

Example 30.2.7. Let $k$ be a field. The ring $R = k[x_1, x_2, x_3, \ldots]/(x_i^2)$ is local with locally nilpotent maximal ideal $\mathfrak m$. There exists no element of $R$ which has annihilator $\mathfrak m$. Hence $\text{Ass}(R) = \emptyset$, and $X = \mathop{\rm Spec}(R)$ is an example of a scheme which has no associated points.

Lemma 30.2.8. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. If $\text{Ass}(\mathcal{F}) \subset U \subset X$ is open, 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 $\text{Supp}(\mathcal{F}') \cap U = \emptyset$. On the other hand, $\text{Ass}(\mathcal{F}') \subset \text{Ass}(\mathcal{F})$ by Lemma 30.2.4. Since also $\text{Ass}(\mathcal{F}') \subset \text{Supp}(\mathcal{F}')$ (Lemma 30.2.3) we conclude $\text{Ass}(\mathcal{F}') = \emptyset$. Hence $\mathcal{F}' = 0$ by Lemma 30.2.6. $\square$

Lemma 30.2.9. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $x \in \text{Supp}(\mathcal{F})$ be a point in the support of $\mathcal{F}$ which is not a specialization of another point of $\text{Supp}(\mathcal{F})$. Then $x \in \text{Ass}(\mathcal{F})$. In particular, any generic point of an irreducible component of $X$ is an associated point of $X$.

Proof. Since $x \in \text{Supp}(\mathcal{F})$ the module $\mathcal{F}_x$ is not zero. Hence $\text{Ass}(\mathcal{F}_x) \subset \mathop{\rm Spec}(\mathcal{O}_{X, x})$ is nonempty by Algebra, Lemma 10.62.7. On the other hand, by assumption $\text{Supp}(\mathcal{F}_x) = \{\mathfrak m_x\}$. Since $\text{Ass}(\mathcal{F}_x) \subset \text{Supp}(\mathcal{F}_x)$ (Algebra, Lemma 10.62.2) we see that $\mathfrak m_x$ is associated to $\mathcal{F}_x$ and we win. $\square$

The following lemma is the analogue of More on Algebra, Lemma 15.21.10.

Lemma 30.2.10. Let $X$ be a locally Noetherian scheme. 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. $\mathcal{F}_x \to \mathcal{G}_x$ is injective, or
  2. $x \not \in \text{Ass}(\mathcal{F})$.

Then $\varphi$ is injective.

Proof. The assumptions imply that $\text{WeakAss}(\text{Ker}(\varphi)) = \emptyset$ and hence $\text{Ker}(\varphi) = 0$ by Lemma 30.2.6. $\square$

Lemma 30.2.11. Let $X$ be a locally Noetherian scheme. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of quasi-coherent $\mathcal{O}_X$-modules. Assume $\mathcal{F}$ is coherent and that for every $x \in X$ one of the following happens

  1. $\mathcal{F}_x \to \mathcal{G}_x$ is an isomorphism, or
  2. $\text{depth}(\mathcal{F}_x) \geq 2$ and $x \not \in \text{Ass}(\mathcal{G})$.

Then $\varphi$ is an isomorphism.

Proof. This is a translation of More on Algebra, Lemma 15.21.11 into the language of schemes. $\square$

    The code snippet corresponding to this tag is a part of the file divisors.tex and is located in lines 26–291 (see updates for more information).

    \section{Associated points}
    \label{section-associated}
    
    \noindent
    Let $R$ be a ring and let $M$ be an $R$-module.
    Recall that a prime $\mathfrak p \subset R$ is {\it associated} to $M$
    if there exists an element of $M$ whose annihilator is $\mathfrak p$.
    See Algebra, Definition \ref{algebra-definition-associated}.
    Here is the definition of associated points
    for quasi-coherent sheaves on schemes
    as given in \cite[IV Definition 3.1.1]{EGA}.
    
    \begin{definition}
    \label{definition-associated}
    Let $X$ be a scheme.
    Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
    \begin{enumerate}
    \item We say $x \in X$ is {\it associated} to $\mathcal{F}$
    if the maximal ideal
    $\mathfrak m_x$ is associated to the $\mathcal{O}_{X, x}$-module
    $\mathcal{F}_x$.
    \item We denote $\text{Ass}(\mathcal{F})$ or $\text{Ass}_X(\mathcal{F})$
    the set of associated points of $\mathcal{F}$.
    \item The {\it associated points of $X$} are the associated
    points of $\mathcal{O}_X$.
    \end{enumerate}
    \end{definition}
    
    \noindent
    These definitions are most useful when $X$ is locally Noetherian
    and $\mathcal{F}$ of finite type.
    For example it may happen that a generic point of an irreducible
    component of $X$ is not associated to $X$, see
    Example \ref{example-no-associated-prime}.
    In the non-Noetherian case it may be more convenient to use weakly
    associated points, see
    Section \ref{section-weakly-associated}.
    Let us link the scheme theoretic notion with the algebraic notion
    on affine opens; note that this correspondence works perfectly only
    for locally Noetherian schemes.
    
    \begin{lemma}
    \label{lemma-associated-affine-open}
    Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
    Let $\Spec(A) = U \subset X$ be an affine open, and set
    $M = \Gamma(U, \mathcal{F})$.
    Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime.
    \begin{enumerate}
    \item If $\mathfrak p$ is associated to $M$, then $x$ is associated
    to $\mathcal{F}$.
    \item If $\mathfrak p$ is finitely generated, then the converse holds
    as well.
    \end{enumerate}
    In particular, if $X$ is locally Noetherian, then the equivalence
    $$
    \mathfrak p \in \text{Ass}(M) \Leftrightarrow x \in \text{Ass}(\mathcal{F})
    $$
    holds for all pairs $(\mathfrak p, x)$ as above.
    \end{lemma}
    
    \begin{proof}
    This follows from
    Algebra, Lemma \ref{algebra-lemma-associated-primes-localize}.
    But we can also argue directly as follows.
    Suppose $\mathfrak p$ is associated to $M$.
    Then there exists an $m \in M$ whose annihilator is $\mathfrak p$.
    Since localization is exact we see that
    $\mathfrak pA_{\mathfrak p}$ is the annihilator of
    $m/1 \in M_{\mathfrak p}$. Since $M_{\mathfrak p} = \mathcal{F}_x$
    (Schemes, Lemma \ref{schemes-lemma-spec-sheaves})
    we conclude that $x$ is associated to $\mathcal{F}$.
    
    \medskip\noindent
    Conversely, assume that $x$ is associated to $\mathcal{F}$,
    and $\mathfrak p$ is finitely generated.
    As $x$ is associated to $\mathcal{F}$
    there exists an element $m' \in M_{\mathfrak p}$ whose
    annihilator is $\mathfrak pA_{\mathfrak p}$. Write
    $m' = m/f$ for some $f \in A$, $f \not \in \mathfrak p$.
    The annihilator $I$ of $m$ is an ideal of $A$ such that
    $IA_{\mathfrak p} = \mathfrak pA_{\mathfrak p}$. Hence
    $I \subset \mathfrak p$, and $(\mathfrak p/I)_{\mathfrak p} = 0$.
    Since $\mathfrak p$ is finitely generated,
    there exists a $g \in A$, $g \not \in \mathfrak p$ such that
    $g(\mathfrak p/I) = 0$. Hence the annihilator of $gm$ is
    $\mathfrak p$ and we win.
    
    \medskip\noindent
    If $X$ is locally Noetherian, then $A$ is Noetherian
    (Properties, Lemma \ref{properties-lemma-locally-Noetherian})
    and $\mathfrak p$ is always finitely generated.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-ass-support}
    Let $X$ be a scheme.
    Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
    Then $\text{Ass}(\mathcal{F}) \subset \text{Supp}(\mathcal{F})$.
    \end{lemma}
    
    \begin{proof}
    This is immediate from the definitions.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-ses-ass}
    Let $X$ be a scheme.
    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
    $\text{Ass}(\mathcal{F}_2) \subset
    \text{Ass}(\mathcal{F}_1) \cup \text{Ass}(\mathcal{F}_3)$
    and
    $\text{Ass}(\mathcal{F}_1) \subset \text{Ass}(\mathcal{F}_2)$.
    \end{lemma}
    
    \begin{proof}
    For every point $x \in X$ the sequence of stalks
    $0 \to \mathcal{F}_{1, x} \to \mathcal{F}_{2, x} \to \mathcal{F}_{3, x} \to 0$
    is a short exact sequence of $\mathcal{O}_{X, x}$-modules.
    Hence the lemma follows from
    Algebra, Lemma \ref{algebra-lemma-ass}.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-finite-ass}
    Let $X$ be a locally Noetherian scheme.
    Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module.
    Then $\text{Ass}(\mathcal{F}) \cap U$ is finite for
    every quasi-compact open $U \subset X$.
    \end{lemma}
    
    \begin{proof}
    This is true because the set of associated primes of a finite module over
    a Noetherian ring is finite, see
    Algebra, Lemma \ref{algebra-lemma-finite-ass}.
    To translate from schemes to algebra use that $U$ is a finite union of
    affine opens, each of these opens is the spectrum of a Noetherian ring
    (Properties, Lemma \ref{properties-lemma-locally-Noetherian}),
    $\mathcal{F}$ corresponds to a finite module over this ring
    (Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-Noetherian}),
    and finally use
    Lemma \ref{lemma-associated-affine-open}.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-ass-zero}
    Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a
    quasi-coherent $\mathcal{O}_X$-module. Then
    $$
    \mathcal{F} = 0 \Leftrightarrow \text{Ass}(\mathcal{F}) = \emptyset.
    $$
    \end{lemma}
    
    \begin{proof}
    If $\mathcal{F} = 0$, then $\text{Ass}(\mathcal{F}) = \emptyset$
    by definition. Conversely, if $\text{Ass}(\mathcal{F}) = \emptyset$,
    then $\mathcal{F} = 0$ by
    Algebra, Lemma \ref{algebra-lemma-ass-zero}.
    To translate from schemes to algebra, restrict to any affine and use
    Lemma \ref{lemma-associated-affine-open}.
    \end{proof}
    
    \begin{example}
    \label{example-no-associated-prime}
    Let $k$ be a field. The ring $R = k[x_1, x_2, x_3, \ldots]/(x_i^2)$
    is local with locally nilpotent maximal ideal $\mathfrak m$.
    There exists no element of $R$ which has annihilator $\mathfrak m$.
    Hence $\text{Ass}(R) = \emptyset$, and $X = \Spec(R)$
    is an example of a scheme which has no associated points.
    \end{example}
    
    \begin{lemma}
    \label{lemma-restriction-injective-open-contains-ass}
    Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a quasi-coherent
    $\mathcal{O}_X$-module. If $\text{Ass}(\mathcal{F}) \subset U \subset X$
    is open, then $\Gamma(X, \mathcal{F}) \to \Gamma(U, \mathcal{F})$
    is injective.
    \end{lemma}
    
    \begin{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
    $\text{Supp}(\mathcal{F}') \cap U = \emptyset$. On the other hand,
    $\text{Ass}(\mathcal{F}') \subset \text{Ass}(\mathcal{F})$
    by Lemma \ref{lemma-ses-ass}. Since also
    $\text{Ass}(\mathcal{F}') \subset \text{Supp}(\mathcal{F}')$
    (Lemma \ref{lemma-ass-support}) we conclude
    $\text{Ass}(\mathcal{F}') = \emptyset$.
    Hence $\mathcal{F}' = 0$ by Lemma \ref{lemma-ass-zero}.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-minimal-support-in-ass}
    Let $X$ be a locally Noetherian scheme.
    Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
    Let $x \in \text{Supp}(\mathcal{F})$ be a point in the support
    of $\mathcal{F}$ which is not a specialization of another point of
    $\text{Supp}(\mathcal{F})$. Then $x \in \text{Ass}(\mathcal{F})$.
    In particular, any generic point of an irreducible component of $X$
    is an associated point of $X$.
    \end{lemma}
    
    \begin{proof}
    Since $x \in \text{Supp}(\mathcal{F})$ the module $\mathcal{F}_x$
    is not zero. Hence
    $\text{Ass}(\mathcal{F}_x) \subset \Spec(\mathcal{O}_{X, x})$
    is nonempty by
    Algebra, Lemma \ref{algebra-lemma-ass-zero}.
    On the other hand, by assumption
    $\text{Supp}(\mathcal{F}_x) = \{\mathfrak m_x\}$.
    Since
    $\text{Ass}(\mathcal{F}_x) \subset \text{Supp}(\mathcal{F}_x)$
    (Algebra, Lemma \ref{algebra-lemma-ass-support})
    we see that $\mathfrak m_x$ is associated to $\mathcal{F}_x$
    and we win.
    \end{proof}
    
    \noindent
    The following lemma is the analogue of
    More on Algebra, Lemma \ref{more-algebra-lemma-check-injective-on-ass}.
    
    \begin{lemma}
    \label{lemma-check-injective-on-ass}
    Let $X$ be a locally Noetherian scheme. 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
    \begin{enumerate}
    \item $\mathcal{F}_x \to \mathcal{G}_x$ is injective, or
    \item $x \not \in \text{Ass}(\mathcal{F})$.
    \end{enumerate}
    Then $\varphi$ is injective.
    \end{lemma}
    
    \begin{proof}
    The assumptions imply that $\text{WeakAss}(\Ker(\varphi)) = \emptyset$
    and hence $\Ker(\varphi) = 0$ by Lemma \ref{lemma-ass-zero}.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-check-isomorphism-via-depth-and-ass}
    Let $X$ be a locally Noetherian scheme. Let
    $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of
    quasi-coherent $\mathcal{O}_X$-modules. Assume $\mathcal{F}$ is coherent
    and that for every $x \in X$ one of the following happens
    \begin{enumerate}
    \item $\mathcal{F}_x \to \mathcal{G}_x$ is an isomorphism, or
    \item $\text{depth}(\mathcal{F}_x) \geq 2$ and
    $x \not \in \text{Ass}(\mathcal{G})$.
    \end{enumerate}
    Then $\varphi$ is an isomorphism.
    \end{lemma}
    
    \begin{proof}
    This is a translation of More on Algebra, Lemma
    \ref{more-algebra-lemma-check-isomorphism-via-depth-and-ass}
    into the language of schemes.
    \end{proof}

    Comments (1)

    Comment #2439 by David Hansen on February 26, 2017 a 12:23 am UTC

    The proof of Lemma 30.2.10 should say "Ass" not "WeakAss".

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