A homological approach to (Grothendieck's) completeness problem for regular LB-spaces

TL;DR

This paper uses homological methods to investigate Grothendieck's longstanding question on the completeness of regular LB-spaces, providing evidence that these spaces may indeed be complete.

Contribution

It establishes the equivalence of derived categories of complete and regular LB-spaces, suggesting they share the same homological properties and supporting an affirmative answer to the completeness problem.

Findings

Proved a canonical triangle functor between the categories is an equivalence.

Demonstrated that the derived categories are well-defined with respect to several exact structures.

Provided evidence favoring the completeness of regular LB-spaces based on homological algebra.

Abstract

We consider the long-standing question of whether every regular LB-space is complete. This problem has been open since the 1950s and originates in Grothendieck's early work in functional analysis. Rather than seeking a direct proof or counterexample, our approach is to study weak versions of the problem using homological methods. We consider the categories of complete and, respectively, regular LB-spaces, establish that their derived categories are well-defined with respect to several exact structures, and show that there are canonical triangle functors between them. If one of these functors were not an equivalence, this would provide a negative answer to Grothendieck's question. In contrast, we prove that one of them is an equivalence. This may be interpreted as evidence in favor of an affirmative answer to the original problem, and it shows in particular that the two classes of spaces share the same homological algebra, even if they were to differ.