29.26 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 29.26.1. Let $X$ be a scheme. The rule which associates to a closed subscheme of $X$ its underlying closed subset defines a bijection
\[ \left\{ \begin{matrix} \text{closed subschemes }Z \subset X
\\ \text{such that }Z \to X\text{ is flat}
\end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{closed subsets }Z \subset X
\\ \text{closed under generalizations}
\end{matrix} \right\} \]
If $Z \subset X$ is such a closed subscheme, every morphism of schemes $g : Y \to X$ with $g(Y) \subset Z$ set theoretically factors (scheme theoretically) through $Z$.
Proof.
The affine case of the bijection is Algebra, Lemma 10.108.4. For general schemes $X$ the bijection follows by covering $X$ by affines and glueing. Details omitted. For the final assertion, observe that the projection $Z \times _{X, g} Y \to Y$ is a flat (Lemma 29.25.8) closed immersion which is bijective on underlying topological spaces and hence must be an isomorphism by the bijection esthablished in the first part of the proof.
$\square$
Lemma 29.26.2. A flat closed immersion of finite presentation is the open immersion of an open and closed subscheme.
Proof.
The affine case is Algebra, Lemma 10.108.5. 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 29.26.3. 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 29.26.1.
It turns out that we can determine when every finite flat $\mathcal{O}_ X$-module is finite locally free using the previous lemma.
Lemma 29.26.4. Let $X$ be a scheme. The following are equivalent
every finite flat quasi-coherent $\mathcal{O}_ X$-module is finite locally free, and
every closed subset $Z \subset X$ which is closed under generalizations is open.
Proof.
In the affine case this is Algebra, Lemma 10.108.6. 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 29.26.1.
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 10.78.5 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 17.21. 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 10.20.1 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 29.26.1. 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.
$\square$
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