Big pure projective modules over commutative noetherian rings: Comparison with the completion

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F25%3A10510378" target="_blank" >RIV/00216208:11320/25:10510378 - isvavai.cz</a>

Result on the web

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=5L7muCsKiP" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=5L7muCsKiP</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1515/forum-2024-0031" target="_blank" >10.1515/forum-2024-0031</a>

Result language

angličtina

Original language name

Big pure projective modules over commutative noetherian rings: Comparison with the completion

Original language description

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 circle times(R) (R) over cap), this allows us to make computations with examples and to prove some realization results.

Czech name

—

Czech description

—

Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10101 - Pure mathematics

Project

<a href="/en/project/GBP201%2F12%2FG028" target="_blank" >GBP201/12/G028: Eduard Čech Institute for algebra, geometry and mathematical physics</a><br>

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2025

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Forum Mathematicum

ISSN

0933-7741

e-ISSN

1435-5337

Volume of the periodical

37

Issue of the periodical within the volume

4

Country of publishing house

DE - GERMANY

Number of pages

44

Pages from-to

1103-1146

UT code for WoS article

001306446100001

EID of the result in the Scopus database

2-s2.0-85203530017