TY - JOUR
T1 - Big pure projective modules over commutative noetherian rings
T2 - Comparison with the completion
AU - Herbera, Dolors
AU - Příhoda, Pavel
AU - Wiegand, Roger
N1 - Publisher Copyright:
© 2024 Walter de Gruyter GmbH, Berlin/Boston 2024.
PY - 2024/9/3
Y1 - 2024/9/3
N2 - 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.
AB - 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.
KW - direct sum decomposition
KW - monoids of modules
KW - Noetherian ring
KW - torsion free modules
KW - trace ideals
KW - Trace ideals
KW - Monoids of modules
KW - Torsion free modules
KW - Direct sum decomposition
UR - http://www.scopus.com/inward/record.url?scp=85203530017&partnerID=8YFLogxK
UR - https://www.mendeley.com/catalogue/68ce85e5-cc84-3bb5-9300-1d96645bdca8/
U2 - 10.1515/forum-2024-0031
DO - 10.1515/forum-2024-0031
M3 - Article
AN - SCOPUS:85203530017
SN - 0933-7741
JO - Forum Mathematicum
JF - Forum Mathematicum
ER -