Torsion-free modules over commutative domains of Krull dimension one
Álvarez Arias, Román (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Herbera i Espinal, Dolors (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Prihoda, Pavel (Charles University. Department of Algebra)

Data:	2025
Descripció:	64 pàg.
Resum:	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.
Ajuts:	Generalitat de Catalunya 2021/FI-B00913 Agencia Estatal de Investigación CEX2020-001084-M Agencia Estatal de Investigación PID2020-113047GB-I00 Generalitat de Catalunya 2021/SGR-01015
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Llengua:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Matèria:	Torsion-free modules ; H-local domain ; Infinite direct sum decomposition ; 2-generated ideals ; Stable categories ; Relatively big projective modules
Publicat a:	Revista Matemática Iberoamericana, Vol. 42, Num. 1 (2026) , p. 123-186, ISSN 0213-2230

DOI: 10.4171/rmi/1564

Postprint 62 p, 628.7 KB

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