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Towards solid abelian groups: A formal proof of Nöbeling's theorem

Dagur Asgeirsson

Abstract

Condensed mathematics, developed by Clausen and Scholze over the last few years, is a new way of studying the interplay between algebra and geometry. It replaces the concept of a topological space by a more sophisticated but better-behaved idea, namely that of a condensed set. Central to the theory are solid abelian groups and liquid vector spaces, analogues of complete topological groups. Nöbeling's theorem, a surprising result from the 1960s about the structure of the abelian group of continuous maps from a profinite space to the integers, is a crucial ingredient in the theory of solid abelian groups; without it one cannot give any nonzero examples of solid abelian groups. We discuss a recently completed formalisation of this result in the Lean theorem prover, and give a more detailed proof than those previously available in the literature. The proof is somewhat unusual in that it requires induction over ordinals -- a technique which has not previously been used to a great extent in formalised mathematics.

Towards solid abelian groups: A formal proof of Nöbeling's theorem

Abstract

Condensed mathematics, developed by Clausen and Scholze over the last few years, is a new way of studying the interplay between algebra and geometry. It replaces the concept of a topological space by a more sophisticated but better-behaved idea, namely that of a condensed set. Central to the theory are solid abelian groups and liquid vector spaces, analogues of complete topological groups. Nöbeling's theorem, a surprising result from the 1960s about the structure of the abelian group of continuous maps from a profinite space to the integers, is a crucial ingredient in the theory of solid abelian groups; without it one cannot give any nonzero examples of solid abelian groups. We discuss a recently completed formalisation of this result in the Lean theorem prover, and give a more detailed proof than those previously available in the literature. The proof is somewhat unusual in that it requires induction over ordinals -- a technique which has not previously been used to a great extent in formalised mathematics.
Paper Structure (21 sections, 26 theorems, 20 equations)

This paper contains 21 sections, 26 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Preliminaries
  4. Order theory
  5. Linear Independence
  6. Cantor's intersection theorem
  7. Cofiltered limits of profinite spaces
  8. The theorem
  9. Notation and generalities
  10. Span
  11. Linear independence
  12. Limit case
  13. Successor case
  14. The formalisation
  15. The implementation of $P$ and $E(S)$
  16. ...and 6 more sections

Key Result

Lemma 3

https://github.com/leanprover-community/mathlib4/blob/ba9f2e5baab51310883778e1ea3b48772581521c/Mathlib/Order/Directed.lean#L174-L177 A monotone map on a poset with a join operation (i.e. a least upper bound of two elements) is directed.

Theorems & Definitions (39)

  • Definition 1
  • Definition 2
  • Lemma 3
  • Remark 4
  • Definition 5
  • Remark 6
  • Remark 7
  • Lemma 8
  • Lemma 9
  • Theorem 10
  • ...and 29 more