Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On cohomology of locally profinite sets

Ko Aoki

Abstract

We construct a locally profinite set of cardinality $\aleph_ω$ 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 $\aleph_ω$ within Zermelo--Fraenkel set theory.

On cohomology of locally profinite sets

Abstract

We construct a locally profinite set of cardinality 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 within Zermelo--Fraenkel set theory.

Paper Structure

This paper contains 9 sections, 19 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Cohomology of locally profinite sets
  3. Construction
  4. Nontriviality
  5. Non-nilpotence
  6. Applications
  7. Boolean rings can have infinite self-injective dimensions
  8. Faithful flatness does not imply descendability
  9. The higher sheaves functor is not (heavy) condensed

Key Result

Theorem A

For $n\geq0$, there is a locally profinite set $U$ of cardinality $\aleph_{n}$ with $H^{n}(U;\mathbf{F}_{2})\neq0$.

Theorems & Definitions (53)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Remark 1.1
  • Theorem E
  • Remark 1.3
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • ...and 43 more