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arXiv · 2310.02716

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

Mishel Carelli·Sergei O. Ivanov
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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.

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Mishel Carelli, Sergei O. Ivanov. 2023-10-04. Transfinite version of the Mittag-Leffler condition for the vanishing of the derived limit. https://arxiv.org/abs/2310.02716

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