Indivisible Sequences and Descendability

TL;DR

This paper introduces indivisible sequences and demonstrates that certain faithfully flat ring maps associated with them can be non-descendable, providing new examples in the context of ring theory.

Contribution

It establishes a novel connection between indivisible sequences and the existence of non-descendable faithfully flat ring maps, including the first such examples between certain countable rings.

Findings

Existence of non-descendable faithfully flat ring maps from indivisible sequences.

First example of a faithfully flat ring map between l_{n-1}-countable rings with a specific descendability exponent.

Counterexamples involving l_{\u221e}-countable rings that are not descendable.

Abstract

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