On cohomology of locally profinite sets

We construct a locally profinite set of cardinality ${\mathrm{\aleph }}_{\omega }$ 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 ${\mathrm{\aleph }}_{\omega }$ within Zermelo–Fraenkel set theory.