Indivisible Sequences and Descendability

We introduce the notion of indivisible sequences and show that to any indivisible sequence $\{S,\mathrm{\Psi }:S\to R\}$ we can associate faithfully flat ring maps $R\to R^{\mathrm{\prime }}$ that are not descendable. As a corollary, we obtain the first example of a faithfully flat ring map between ${\mathrm{\aleph }}_{n-1}$-countable rings that has descendability exponent $n$, and indeed a faithfully flat ring map between ${\mathrm{\aleph }}_{\omega }$-countable rings that is not descendable.