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Tag 04PV

28.25. Flat closed immersions

Connected components of schemes are not always open. But they do always have a canonical scheme structure. We explain this in this section.

Lemma 28.25.1. Let $X$ be a scheme. The rule which associates to a closed subscheme of $X$ its underlying closed subset defines a bijection $$ \left\{ \begin{matrix} \text{closed subschemes }Z \subset X \\ \text{such that }Z \to X\text{ is flat} \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{closed subsets }Z \subset X \\ \text{closed under generalizations} \end{matrix} \right\} $$

Proof. The affine case is Algebra, Lemma 10.107.4. In general the lemma follows by covering $X$ by affines and glueing. Details omitted. $\square$

Lemma 28.25.2. A flat closed immersion of finite presentation is the open immersion of an open and closed subscheme.

Proof. The affine case is Algebra, Lemma 10.107.5. In general the lemma follows by covering $X$ by affines. Details omitted. $\square$

Note that a connected component $T$ of a scheme $X$ is a closed subset stable under generalization. Hence the following definition makes sense.

Definition 28.25.3. Let $X$ be a scheme. Let $T \subset X$ be a connected component. The canonical scheme structure on $T$ is the unique scheme structure on $T$ such that the closed immersion $T \to X$ is flat, see Lemma 28.25.1.

It turns out that we can determine when every finite flat $\mathcal{O}_X$-module is finite locally free using the previous lemma.

Lemma 28.25.4. Let $X$ be a scheme. The following are equivalent

  1. every finite flat quasi-coherent $\mathcal{O}_X$-module is finite locally free, and
  2. every closed subset $Z \subset X$ which is closed under generalizations is open.

Proof. In the affine case this is Algebra, Lemma 10.107.6. The scheme case does not follow directly from the affine case, so we simply repeat the arguments.

Assume (1). Consider a closed immersion $i : Z \to X$ such that $i$ is flat. Then $i_*\mathcal{O}_Z$ is quasi-coherent and flat, hence finite locally free by (1). Thus $Z = \text{Supp}(i_*\mathcal{O}_Z)$ is also open and we see that (2) holds. Hence the implication (1) $\Rightarrow$ (2) follows from the characterization of flat closed immersions in Lemma 28.25.1.

For the converse assume that $X$ satisfies (2). Let $\mathcal{F}$ be a finite flat quasi-coherent $\mathcal{O}_X$-module. The support $Z = \text{Supp}(\mathcal{F})$ of $\mathcal{F}$ is closed, see Modules, Lemma 17.9.6. On the other hand, if $x \leadsto x'$ is a specialization, then by Algebra, Lemma 10.77.4 the module $\mathcal{F}_{x'}$ is free over $\mathcal{O}_{X, x'}$, and $$ \mathcal{F}_x = \mathcal{F}_{x'} \otimes_{\mathcal{O}_{X, x'}} \mathcal{O}_{X, x}. $$ Hence $x' \in \text{Supp}(\mathcal{F}) \Rightarrow x \in \text{Supp}(\mathcal{F})$, in other words, the support is closed under generalization. As $X$ satisfies (2) we see that the support of $\mathcal{F}$ is open and closed. The modules $\wedge^i(\mathcal{F})$, $i = 1, 2, 3, \ldots$ are finite flat quasi-coherent $\mathcal{O}_X$-modules also, see Modules, Section 17.19. Note that $\text{Supp}(\wedge^{i + 1}(\mathcal{F})) \subset \text{Supp}(\wedge^i(\mathcal{F}))$. Thus we see that there exists a decomposition $$ X = U_0 \amalg U_1 \amalg U_2 \amalg \ldots $$ by open and closed subsets such that the support of $\wedge^i(\mathcal{F})$ is $U_i \cup U_{i + 1} \cup \ldots$ for all $i$. Let $x$ be a point of $X$, and say $x \in U_r$. Note that $\wedge^i(\mathcal{F})_x \otimes \kappa(x) = \wedge^i(\mathcal{F}_x \otimes \kappa(x))$. Hence, $x \in U_r$ implies that $\mathcal{F}_x \otimes \kappa(x)$ is a vector space of dimension $r$. By Nakayama's lemma, see Algebra, Lemma 10.19.1 we can choose an affine open neighbourhood $U \subset U_r \subset X$ of $x$ and sections $s_1, \ldots, s_r \in \mathcal{F}(U)$ such that the induced map $$ \mathcal{O}_U^{\oplus r} \longrightarrow \mathcal{F}|_U, \quad (f_1, \ldots, f_r) \longmapsto \sum f_i s_i $$ is surjective. This means that $\wedge^r(\mathcal{F}|_U)$ is a finite flat quasi-coherent $\mathcal{O}_U$-module whose support is all of $U$. By the above it is generated by a single element, namely $s_1 \wedge \ldots \wedge s_r$. Hence $\wedge^r(\mathcal{F}|_U) \cong \mathcal{O}_U/\mathcal{I}$ for some quasi-coherent sheaf of ideals $\mathcal{I}$ such that $\mathcal{O}_U/\mathcal{I}$ is flat over $\mathcal{O}_U$ and such that $V(\mathcal{I}) = U$. It follows that $\mathcal{I} = 0$ by applying Lemma 28.25.1. Thus $s_1 \wedge \ldots \wedge s_r$ is a basis for $\wedge^r(\mathcal{F}|_U)$ and it follows that the displayed map is injective as well as surjective. This proves that $\mathcal{F}$ is finite locally free as desired. $\square$

    The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 4524–4679 (see updates for more information).

    \section{Flat closed immersions}
    \label{section-flat-closed-immersions}
    
    \noindent
    Connected components of schemes are not always open. But they do always
    have a canonical scheme structure. We explain this in this section.
    
    \begin{lemma}
    \label{lemma-characterize-flat-closed-immersions}
    Let $X$ be a scheme. The rule which associates to a closed subscheme
    of $X$ its underlying closed subset defines a bijection
    $$
    \left\{
    \begin{matrix}
    \text{closed subschemes }Z \subset X \\
    \text{such that }Z \to X\text{ is flat}
    \end{matrix}
    \right\}
    \leftrightarrow
    \left\{
    \begin{matrix}
    \text{closed subsets }Z \subset X \\
    \text{closed under generalizations}
    \end{matrix}
    \right\}
    $$
    \end{lemma}
    
    \begin{proof}
    The affine case is
    Algebra, Lemma \ref{algebra-lemma-pure-open-closed-specializations}.
    In general the lemma follows by covering $X$ by affines and glueing.
    Details omitted.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-flat-closed-immersions-finite-presentation}
    A flat closed immersion of finite presentation
    is the open immersion of an open and closed subscheme.
    \end{lemma}
    
    \begin{proof}
    The affine case is
    Algebra, Lemma \ref{algebra-lemma-finitely-generated-pure-ideal}.
    In general the lemma follows by covering $X$ by affines.
    Details omitted.
    \end{proof}
    
    \noindent
    Note that a connected component $T$ of a scheme $X$ is a closed
    subset stable under generalization. Hence the following definition
    makes sense.
    
    \begin{definition}
    \label{definition-scheme-structure-connected-component}
    Let $X$ be a scheme. Let $T \subset X$ be a connected component.
    The {\it canonical scheme structure on $T$} is the unique
    scheme structure on $T$ such that the closed immersion $T \to X$
    is flat, see
    Lemma \ref{lemma-characterize-flat-closed-immersions}.
    \end{definition}
    
    \noindent
    It turns out that we can determine when every finite flat
    $\mathcal{O}_X$-module is finite locally free using the previous lemma.
    
    \begin{lemma}
    \label{lemma-finite-flat-is-finite-locally-free}
    Let $X$ be a scheme. The following are equivalent
    \begin{enumerate}
    \item every finite flat quasi-coherent $\mathcal{O}_X$-module is
    finite locally free, and
    \item every closed subset $Z \subset X$ which is closed under generalizations
    is open.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    In the affine case this is
    Algebra, Lemma \ref{algebra-lemma-finite-flat-module-finitely-presented}.
    The scheme case does not follow directly from the affine case, so we
    simply repeat the arguments.
    
    \medskip\noindent
    Assume (1). Consider a closed immersion $i : Z \to X$ such that $i$ is flat.
    Then $i_*\mathcal{O}_Z$ is quasi-coherent and flat, hence finite locally
    free by (1). Thus $Z = \text{Supp}(i_*\mathcal{O}_Z)$ is also open and we see
    that (2) holds. Hence the implication (1) $\Rightarrow$ (2) follows from
    the characterization of flat closed immersions in
    Lemma \ref{lemma-characterize-flat-closed-immersions}.
    
    \medskip\noindent
    For the converse assume that $X$ satisfies (2).
    Let $\mathcal{F}$ be a finite flat quasi-coherent $\mathcal{O}_X$-module.
    The support $Z = \text{Supp}(\mathcal{F})$ of $\mathcal{F}$ is closed, see
    Modules, Lemma \ref{modules-lemma-support-finite-type-closed}.
    On the other hand, if $x \leadsto x'$ is a specialization, then by
    Algebra, Lemma \ref{algebra-lemma-finite-flat-local}
    the module $\mathcal{F}_{x'}$ is free over $\mathcal{O}_{X, x'}$, and
    $$
    \mathcal{F}_x =
    \mathcal{F}_{x'} \otimes_{\mathcal{O}_{X, x'}} \mathcal{O}_{X, x}.
    $$
    Hence
    $x' \in \text{Supp}(\mathcal{F}) \Rightarrow x \in \text{Supp}(\mathcal{F})$,
    in other words, the support is closed under generalization.
    As $X$ satisfies (2) we see that the support of $\mathcal{F}$
    is open and closed. The modules $\wedge^i(\mathcal{F})$, $i = 1, 2, 3, \ldots$
    are finite flat quasi-coherent $\mathcal{O}_X$-modules
    also, see
    Modules, Section \ref{modules-section-symmetric-exterior}.
    Note that
    $\text{Supp}(\wedge^{i + 1}(\mathcal{F})) \subset
    \text{Supp}(\wedge^i(\mathcal{F}))$.
    Thus we see that there exists a decomposition
    $$
    X = U_0 \amalg U_1 \amalg U_2 \amalg \ldots
    $$
    by open and closed subsets such that the support of
    $\wedge^i(\mathcal{F})$ is $U_i \cup U_{i + 1} \cup \ldots$ for all $i$.
    Let $x$ be a point of $X$, and say $x \in U_r$.
    Note that
    $\wedge^i(\mathcal{F})_x \otimes \kappa(x) =
    \wedge^i(\mathcal{F}_x \otimes \kappa(x))$.
    Hence, $x \in U_r$ implies that $\mathcal{F}_x \otimes \kappa(x)$
    is a vector space of dimension $r$. By Nakayama's lemma, see
    Algebra, Lemma \ref{algebra-lemma-NAK}
    we can choose an affine open neighbourhood $U \subset U_r \subset X$
    of $x$ and sections $s_1, \ldots, s_r \in \mathcal{F}(U)$ such that
    the induced map
    $$
    \mathcal{O}_U^{\oplus r} \longrightarrow \mathcal{F}|_U, \quad
    (f_1, \ldots, f_r) \longmapsto \sum f_i s_i
    $$
    is surjective. This means that
    $\wedge^r(\mathcal{F}|_U)$ is a finite flat quasi-coherent
    $\mathcal{O}_U$-module whose support is all of $U$.
    By the above it is generated by a single element, namely
    $s_1 \wedge \ldots \wedge s_r$. Hence
    $\wedge^r(\mathcal{F}|_U) \cong \mathcal{O}_U/\mathcal{I}$
    for some quasi-coherent sheaf of ideals $\mathcal{I}$
    such that $\mathcal{O}_U/\mathcal{I}$ is flat over $\mathcal{O}_U$ and
    such that $V(\mathcal{I}) = U$.
    It follows that $\mathcal{I} = 0$ by applying
    Lemma \ref{lemma-characterize-flat-closed-immersions}.
    Thus $s_1 \wedge \ldots \wedge s_r$ is a basis for
    $\wedge^r(\mathcal{F}|_U)$ and it follows that the displayed map is injective
    as well as surjective. This proves that $\mathcal{F}$ is finite locally free
    as desired.
    \end{proof}

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