Transfinite version of the Mittag-Leffler condition for the vanishing of the derived limit

TL;DR

This paper establishes a transfinite generalization of the Mittag-Leffler condition, providing a necessary and sufficient criterion for the vanishing of the first derived limit in inverse sequences within certain abelian categories.

Contribution

It introduces a transfinite version of the Mittag-Leffler condition and characterizes inverse sequences with vanishing limits and derived limits in broad abelian categories.

Findings

Provides a necessary and sufficient condition for ${ m lim}^1 S=0$.

Characterizes the class of inverse sequences with ${ m lim} S=0$ and ${ m lim}^1 S=0$.

Extends classical results to transfinite and categorical contexts.

Abstract

We give a necessary and sufficient condition for an inverse sequence $S_0 \leftarrow S_1 \leftarrow \dots$ indexed by natural numbers to have ${\rm lim}^1S=0$. This condition can be treated as a transfinite version of the Mittag-Leffler condition. We consider inverse sequences in an arbitrary abelian category having a generator and satisfying Grothendieck axioms ${\rm (AB3)}$ and ${\rm (AB4^*)}.$ We also show that the class of inverse sequences $S$ such that ${\rm lim}\: S={\rm lim}^1 S=0$ is the least class of inverse sequences containing the trivial inverse sequence and closed with respect to small limits and a certain type of extensions.