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

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

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

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GBP201%2F12%2FG028" target="_blank" >GBP201/12/G028: Ústav Eduarda Čecha pro algebru, geometrii a matematickou fyziku</a><br>

Návaznosti

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

Rok uplatnění

2025

Kód důvěrnosti údajů

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

Název periodika

Forum Mathematicum

ISSN

0933-7741

e-ISSN

1435-5337

Svazek periodika

37

Číslo periodika v rámci svazku

4

Stát vydavatele periodika

DE - Spolková republika Německo

Počet stran výsledku

44

Strana od-do

1103-1146

Kód UT WoS článku

001306446100001

EID výsledku v databázi Scopus

2-s2.0-85203530017