Article
| Title: | Mittag-Leffler modules and definable subcategories II (English) |
| Author: | Rothmaler, Philipp |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 75 |
| Issue: | 4 |
| Year: | 2025 |
| Pages: | 1105-1116 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | A countably generated module is Mittag-Leffler if and only if it is pure-projective, i.e., a direct summand of a direct sum of finitely presented modules. Trying to generalize this description to countably generated relative Mittag-Leffler modules, one runs into serious obstacles. The last theorem of Part I of the same title describes them as what was called uniform relative pure epimorphic images of a specific kind of $\omega $-limits of finitely presented modules. A second part of this theorem attempted to make the $\omega $-limits in question more concrete. In the formulation the application of those uniform pure epimorphisms was erroneously omitted. Correcting this led to a better result to be presented here. It states that under a mild assumption on the context, every countably generated relative Mittag-Leffler module `in the context' is a direct summand of a certain preenvelope of a union of a relatively pure $\omega $-chain of finitely presented modules. In conclusion a number of examples are presented that start with and grew out of the study of $\cal L$-purity of monomorphisms in $\mathbb {Z}$-Mod for $\cal L$, the definable subcategory of divisible Abelian groups. Rings that get particular attention in this are RD-rings. (English) |
| Keyword: | countably generated relative Mittag-Leffler |
| Keyword: | atomic |
| Keyword: | pure-projective module |
| Keyword: | definable subcategory |
| Keyword: | divisible module over RD-domains |
| Keyword: | preenvelope |
| Keyword: | coherent |
| Keyword: | Noetherian |
| Keyword: | v.N. regular |
| Keyword: | RD-purity |
| MSC: | 16B70 |
| MSC: | 16D40 |
| MSC: | 16D70 |
| MSC: | 16D80 |
| MSC: | 16S90 |
| DOI: | 10.21136/CMJ.2025.0293-24 |
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| Date available: | 2025-12-20T07:08:16Z |
| Last updated: | 2025-12-22 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153233 |
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