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 $Z$.

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

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

**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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