On cohomology of locally profinite sets

TL;DR

This paper constructs a complex locally profinite set with unique cohomological properties and presents the first example of a nondescendable faithfully flat map between large commutative rings within standard set theory.

Contribution

It introduces a novel locally profinite set with infinite cohomology classes and constructs a groundbreaking nondescendable faithfully flat ring map of size .

Findings

Constructed a locally profinite set of size  with infinitely many non-vanishing first cohomology classes.

Provided the first example of a nondescendable faithfully flat map between large commutative rings in ZF set theory.

Abstract

We construct a locally profinite set of cardinality $\aleph_{\omega}$ with infinitely many first cohomology classes of which any distinct finite product does not vanish. Building on this, we construct the first example of a nondescendable faithfully flat map between commutative rings of cardinality $\aleph_{\omega}$ within Zermelo--Fraenkel set theory.