The Stacks project

Remark 3.9.10. Let $R$ be a ring. Suppose we consider the ring $\prod _{\mathfrak p \in \mathop{\mathrm{Spec}}(R)} \kappa (\mathfrak p)$. The cardinality of this ring is bounded by $|R|^{2^{|R|}}$, but is not bounded by $|R|^{\aleph _0}$ in general. For example if $R = \mathbf{C}[x]$ it is not bounded by $|R|^{\aleph _0}$ and if $R = \prod _{n \in \mathbf{N}} \mathbf{F}_2$ it is not bounded by $|R|^{|R|}$. Thus the “And so on” of Lemma 3.9.9 above should be taken with a grain of salt. Of course, if it ever becomes necessary to consider these rings in arguments pertaining to fppf/étale cohomology, then we can change the function $Bound$ above into the function $\kappa \mapsto \kappa ^{2^\kappa }$.