Torsion-free modules over commutative domains of Krull dimension one
Let R be a domain of Krull dimension one. We study when the class F of modules over R that are arbitrary direct sums of finitely generated torsion-free modules is closed under direct summands. If R is local, we show that F is closed under direct summands if and only if any indecomposable, finitely g...
| Authors: | ; ; |
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| Format: | Article |
| Publication date: | 2026 |
| Country: | España |
| Institution: | Universitat Autònoma de Barcelona |
| Repository: | Dipòsit Digital de Documents de la UAB |
| Language: | English |
| OAI Identifier: | oai:ddd.uab.cat:325852 |
| Online Access: | https://ddd.uab.cat/record/325852 https://dx.doi.org/urn:doi:10.4171/rmi/1564 |
| Access Level: | Open access |
| Keyword: | Torsion-free modules H-local domain Infinite direct sum decomposition 2-generated ideals Stable categories Relatively big projective modules |
| Summary: | Let R be a domain of Krull dimension one. We study when the class F of modules over R that are arbitrary direct sums of finitely generated torsion-free modules is closed under direct summands. If R is local, we show that F is closed under direct summands if and only if any indecomposable, finitely generated, torsion-free module has local endomorphism ring. If, in addition, R is noetherian, this is equivalent to saying that the normalization of R is a local ring. If R is an h-local domain of Krull dimension 1 and F R is closed under direct summands, then the property is inherited by the localizations of R at maximal ideals. Moreover, any localization of R at a maximal ideal, except maybe one, satisfies that any finitely generated ideal is 2-generated. The converse is true when the domain R is, in addition, integrally closed, or noetherian semilocal, or noetherian with module-finite normalization. Finally, over a commutative domain of finite character and with no restriction on the Krull dimension, we show that the isomorphism classes of countably generated modules in F are determined by their genus. |
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