The Stacks project

Remark 61.11.4. Let $A$ be a ring. Let $\kappa $ be an infinite cardinal bigger or equal than the cardinality of $A$. Then the cardinality of the ring $D$ constructed in Proposition 61.11.3 is at most

\[ \kappa ^{2^{2^{2^\kappa }}}. \]

Namely, the ring map $A \to D$ is constructed as a composition

\[ A \to A_ w = A' \to C' \to C \to D. \]

Here the first three steps of the construction are carried out in the first paragraph of the proof of Lemma 61.8.7. For the first step we have $|A_ w| \leq \kappa $ by Remark 61.5.4. We have $|C'| \leq \kappa $ by Remark 61.8.2. Then $|C| \leq \kappa $ because $C$ is a localization of $(C')_ w$ (it is constructed from $C'$ by an application of Lemma 61.5.7 in the proof of Lemma 61.5.8). Thus $C$ has at most $2^\kappa $ maximal ideals. Finally, the ring map $C \to D$ identifies local rings and the cardinality of the set of maximal ideals of $D$ is at most $2^{2^{2^\kappa }}$ by Topology, Remark 5.26.10. Since $D \subset \prod _{\mathfrak m \subset D} D_\mathfrak m$ we see that $D$ has at most the size displayed above.

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