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Tag 05JU

Chapter 28: Morphisms of Schemes > Section 28.5: Supports of modules

Lemma 28.5.4. Let $\mathcal{F}$ be a finite type quasi-coherent module on a scheme $X$. There exists a smallest closed subscheme $i : Z \to X$ such that there exists a quasi-coherent $\mathcal{O}_Z$-module $\mathcal{G}$ with $i_*\mathcal{G} \cong \mathcal{F}$. Moreover:

  1. If $\mathop{\rm Spec}(A) \subset X$ is any affine open, and $\mathcal{F}|_{\mathop{\rm Spec}(A)} = \widetilde{M}$ then $Z \cap \mathop{\rm Spec}(A) = \mathop{\rm Spec}(A/I)$ where $I = \text{Ann}_A(M)$.
  2. The quasi-coherent sheaf $\mathcal{G}$ is unique up to unique isomorphism.
  3. The quasi-coherent sheaf $\mathcal{G}$ is of finite type.
  4. The support of $\mathcal{G}$ and of $\mathcal{F}$ is $Z$.

Proof. Suppose that $i' : Z' \to X$ is a closed subscheme which satisfies the description on open affines from the lemma. Then by Lemma 28.4.1 we see that $\mathcal{F} \cong i'_*\mathcal{G}'$ for some unique quasi-coherent sheaf $\mathcal{G}'$ on $Z'$. Furthermore, it is clear that $Z'$ is the smallest closed subscheme with this property (by the same lemma). Finally, using Properties, Lemma 27.16.1 and Algebra, Lemma 10.5.5 it follows that $\mathcal{G}'$ is of finite type. We have $\text{Supp}(\mathcal{G}') = Z$ by Algebra, Lemma 10.39.5. Hence, in order to prove the lemma it suffices to show that the characterization in (1) actually does define a closed subscheme. And, in order to do this it suffices to prove that the given rule produces a quasi-coherent sheaf of ideals, see Lemma 28.2.3. This comes down to the following algebra fact: If $A$ is a ring, $f \in A$, and $M$ is a finite $A$-module, then $\text{Ann}_A(M)_f = \text{Ann}_{A_f}(M_f)$. We omit the proof. $\square$

    The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 695–711 (see updates for more information).

    \begin{lemma}
    \label{lemma-scheme-theoretic-support}
    Let $\mathcal{F}$ be a finite type quasi-coherent module
    on a scheme $X$. There exists a smallest closed subscheme
    $i : Z \to X$ such that there exists a quasi-coherent
    $\mathcal{O}_Z$-module $\mathcal{G}$ with
    $i_*\mathcal{G} \cong \mathcal{F}$. Moreover:
    \begin{enumerate}
    \item If $\Spec(A) \subset X$ is any affine open, and
    $\mathcal{F}|_{\Spec(A)} = \widetilde{M}$ then
    $Z \cap \Spec(A) = \Spec(A/I)$ where $I = \text{Ann}_A(M)$.
    \item The quasi-coherent sheaf $\mathcal{G}$ is unique up to unique
    isomorphism.
    \item The quasi-coherent sheaf $\mathcal{G}$ is of finite type.
    \item The support of $\mathcal{G}$ and of $\mathcal{F}$ is $Z$.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Suppose that $i' : Z' \to X$ is a closed subscheme which satisfies the
    description on open affines from the lemma. Then by
    Lemma \ref{lemma-i-star-equivalence}
    we see that $\mathcal{F} \cong i'_*\mathcal{G}'$ for some unique
    quasi-coherent sheaf $\mathcal{G}'$ on $Z'$. Furthermore, it is clear
    that $Z'$ is the smallest closed subscheme with this property (by the
    same lemma). Finally, using
    Properties, Lemma \ref{properties-lemma-finite-type-module}
    and
    Algebra, Lemma \ref{algebra-lemma-finite-over-subring}
    it follows that $\mathcal{G}'$ is of finite type. We have
    $\text{Supp}(\mathcal{G}') = Z$ by
    Algebra, Lemma \ref{algebra-lemma-support-closed}.
    Hence, in order to prove the lemma it suffices to show that
    the characterization in (1) actually does define a closed subscheme.
    And, in order to do this it suffices to prove that the given rule
    produces a quasi-coherent sheaf of ideals, see
    Lemma \ref{lemma-closed-immersion-bijection-ideals}.
    This comes down to the following algebra fact: If $A$ is a ring, $f \in A$,
    and $M$ is a finite $A$-module, then
    $\text{Ann}_A(M)_f = \text{Ann}_{A_f}(M_f)$.
    We omit the proof.
    \end{proof}

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