A descriptive approach to higher derived limits

TL;DR

This paper introduces a new complexity measure for elements of higher derived limits over a specific directed set, establishing conditions under which these limits are additive and providing vanishing results.

Contribution

It develops a novel complexity measure for higher derived limits and isolates a partition principle that ensures additivity and simplifies the understanding of these limits.

Findings

Cocycles of bounded complexity are images of cochains of similar complexity.

The partition principle implies the additivity of the derived limit functor.

Vanishing results for higher derived limits are derived from the partition principle.

Abstract

We present a new aspect of the study of higher derived limits. More precisely, we introduce a complexity measure for the elements of higher derived limits over the directed set $\Omega$ of functions from $\mathbb{N}$ to $\mathbb{N}$ and prove that cocycles of this complexity are images of cochains of the roughly the same complexity. In the course of this work, we isolate a partition principle for powers of directed sets and show that whenever this principle holds, the corresponding derived limit $\mathrm{lim}^n$ is additive; vanishing results for this limit are the typical corollary. The formulation of this partition hypothesis synthesizes and clarifies several recent advances in this area.