Big pure projective modules over commutative noetherian rings : Comparison with the completion
Herbera i Espinal, Dolors (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Prihoda, Pavel (Charles University. Department of Algebra)
Wiegand, Roger (University of Nebraska. Department of Mathematics)

Fecha:	2024
Resumen:	A module over a ring R is pure projective provided it is isomorphic to a direct summand of a direct sum of finitely presented modules. We develop tools for the classification of pure projective modules over commutative noetherian rings. In particular, for a fixed finitely presented module M, we consider Add (M), which consists of direct summands of direct sums of copies of M. We are primarily interested in the case where R is a one-dimensional, local domain, and in torsion-free (or Cohen-Macaulay) modules. We show that, even in this case, Add (M) can have an abundance of modules that are not direct sums of finitely generated ones. Our work is based on the fact that such infinitely generated direct summands are all determined by finitely generated data. Namely, idempotent/trace ideals of the endomorphism ring of M and finitely generated projective modules modulo such idempotent ideals. This allows us to extend the classical theory developed to study the behaviour of direct sum decomposition of finitely generated modules comparing with their completion to the infinitely generated case. We study the structure of the monoid V ∗ (M), of isomorphism classes of countably generated modules in Add (M) with the addition induced by the direct sum. We show that V ∗ (M) is a submonoid of V ∗ (M ⊗R R), this allows us to make computations with examples and to prove some realization results.
Ayudas:	Ministerio de Economía y Competitividad MTM2014-53644-P Agencia Estatal de Investigación MTM2017-83487-P Agencia Estatal de Investigación PID2020-113047GB-I00 Generalitat de Catalunya 2021/SGR-01015
Derechos:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, i la comunicació pública de l'obra, sempre que no sigui amb finalitats comercials, i sempre que es reconegui l'autoria de l'obra original. No es permet la creació d'obres derivades.
Lengua:	Anglès
Documento:	Article ; recerca ; Versió acceptada per publicar
Materia:	Direct sum decomposition ; Monoids of modules ; Noetherian ring ; Torsion free modules ; Trace ideals
Publicado en:	Forum Mathematicum, Vol. 37, Num. 4 (September 2024) , p. 1103-1146, ISSN 1435-5337

DOI: 10.1515/forum-2024-0031

Postprint 64 p, 702.3 KB

El registro aparece en las colecciones:
Artículos > Artículos de investigación
Artículos > Artículos publicados