The Stacks project

Lemma 3.9.5. Let $S$ be a scheme. Let $S = \bigcup _{i \in I} S_ i$ be an open covering. Then $\text{size}(S) \leq \max \{ |I|, \sup _ i\{ \text{size}(S_ i)\} \} $.

Proof. Let $U \subset S$ be any affine open. Since $U$ is quasi-compact there exist finitely many elements $i_1, \ldots , i_ n \in I$ and affine opens $U_ i \subset U \cap S_ i$ such that $U = U_1 \cup U_2 \cup \ldots \cup U_ n$. Thus

\[ |\Gamma (U, \mathcal{O}_ U)| \leq |\Gamma (U_1, \mathcal{O})| \otimes \ldots \otimes |\Gamma (U_ n, \mathcal{O})| \leq \sup \nolimits _ i\{ \text{size}(S_ i)\} \]

Moreover, it shows that the set of affine opens of $S$ has cardinality less than or equal to the cardinality of the set

\[ \coprod _{n \in \omega } \coprod _{i_1, \ldots , i_ n \in I} \{ \text{affine opens of }S_{i_1}\} \times \ldots \times \{ \text{affine opens of }S_{i_ n}\} . \]

Each of the sets inside the disjoint union has cardinality at most $\sup _ i\{ \text{size}(S_ i)\} $. The index set has cardinality at most $\max \{ |I|, \aleph _0\} $, see [Ch. I, 10.13, Kunen]. Hence by [Lemma 5.8, Jech] the cardinality of the coproduct is at most $\max \{ \aleph _0, |I|\} \otimes \sup _ i\{ \text{size}(S_ i)\} $. The lemma follows. $\square$


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