Transfinite version of the Mittag-Leffler condition for the vanishing of the derived limit
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912019974717440 |
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| author | Carelli, Mishel Ivanov, Sergei O. |
| author_facet | Carelli, Mishel Ivanov, Sergei O. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_02716 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Transfinite version of the Mittag-Leffler condition for the vanishing of the derived limit Carelli, Mishel Ivanov, Sergei O. K-Theory and Homology 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. |
| title | Transfinite version of the Mittag-Leffler condition for the vanishing of the derived limit |
| topic | K-Theory and Homology |
| url | https://arxiv.org/abs/2310.02716 |