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

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 ${\lim }^{1}S=0$. 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 $(AB3)$ and $(\mathrm{AB}{4}^{\ast }).$ We also show that the class of inverse sequences $S$ such that $\lim S={\lim }^{1}S=0$ 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.