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Statements

Subject Item
n2:RIV%2F00216208%3A11320%2F12%3A10128138%21RIV13-GA0-11320___
rdf:type
n10:Vysledek skos:Concept
rdfs:seeAlso
http://dx.doi.org/10.1515/FORM.2011.101
dcterms:description
Kaplansky classes emerged in the context of Enochs' solution of the Flat Cover Conjecture. Their connection to abstract model theory goes back to Baldwin et al.: a class C of roots of Ext is a Kaplansky class closed under direct limits if and only if the pair (C, {=) is an abstract elementary class (AEC) in the sense of Shelah. We prove that this AEC has finite character in case C = C-perpendicular to' for a class C' of pure-injective modules. In particular, all AECs of roots of Ext over any right noetherian right hereditary ring R have finite character (but the case of general rings remains open). If (C, {=) is an AEC of roots of Ext, then C is known to be a covering class. However, Kaplansky classes need not even be precovering in general: We prove that the class D of all aleph(1)-projective modules (which is equal to the class of all flat Mittag-Leffler modules) is a Kaplansky class for any ring R, but it fails to be precovering in case R is not right perfect, the class (perpendicular to)(D-perpendicular to) equals the class of all flat modules and consists of modules of projective dimension {= 1. Assuming the Singular Cardinal Hypothesis, we prove that D is not precovering for each countable non-right perfect ring R. Kaplansky classes emerged in the context of Enochs' solution of the Flat Cover Conjecture. Their connection to abstract model theory goes back to Baldwin et al.: a class C of roots of Ext is a Kaplansky class closed under direct limits if and only if the pair (C, {=) is an abstract elementary class (AEC) in the sense of Shelah. We prove that this AEC has finite character in case C = C-perpendicular to' for a class C' of pure-injective modules. In particular, all AECs of roots of Ext over any right noetherian right hereditary ring R have finite character (but the case of general rings remains open). If (C, {=) is an AEC of roots of Ext, then C is known to be a covering class. However, Kaplansky classes need not even be precovering in general: We prove that the class D of all aleph(1)-projective modules (which is equal to the class of all flat Mittag-Leffler modules) is a Kaplansky class for any ring R, but it fails to be precovering in case R is not right perfect, the class (perpendicular to)(D-perpendicular to) equals the class of all flat modules and consists of modules of projective dimension {= 1. Assuming the Singular Cardinal Hypothesis, we prove that D is not precovering for each countable non-right perfect ring R.
dcterms:title
Kaplansky classes, finite character and aleph(1)-projectivity Kaplansky classes, finite character and aleph(1)-projectivity
skos:prefLabel
Kaplansky classes, finite character and aleph(1)-projectivity Kaplansky classes, finite character and aleph(1)-projectivity
skos:notation
RIV/00216208:11320/12:10128138!RIV13-GA0-11320___
n10:predkladatel
n20:orjk%3A11320
n3:aktivita
n11:P n11:Z
n3:aktivity
P(GA201/09/0816), P(GD201/09/H012), P(MEB101005), Z(MSM0021620839)
n3:cisloPeriodika
5
n3:dodaniDat
n16:2013
n3:domaciTvurceVysledku
n12:7944578 n12:6265219
n3:druhVysledku
n4:J
n3:duvernostUdaju
n15:S
n3:entitaPredkladatele
n17:predkladatel
n3:idSjednocenehoVysledku
144348
n3:idVysledku
RIV/00216208:11320/12:10128138
n3:jazykVysledku
n21:eng
n3:klicovaSlova
aleph1-projectivity; character; finite; classes; Kaplansky
n3:klicoveSlovo
n5:aleph1-projectivity n5:Kaplansky n5:character n5:finite n5:classes
n3:kodStatuVydavatele
DE - Spolková republika Německo
n3:kontrolniKodProRIV
[DB92A95EE8E5]
n3:nazevZdroje
Forum Mathematicum
n3:obor
n7:BA
n3:pocetDomacichTvurcuVysledku
2
n3:pocetTvurcuVysledku
2
n3:projekt
n13:MEB101005 n13:GA201%2F09%2F0816 n13:GD201%2F09%2FH012
n3:rokUplatneniVysledku
n16:2012
n3:svazekPeriodika
24
n3:tvurceVysledku
Šaroch, Jan Trlifaj, Jan
n3:wos
000309161800008
n3:zamer
n22:MSM0021620839
s:issn
0933-7741
s:numberOfPages
19
n19:doi
10.1515/FORM.2011.101
n8:organizacniJednotka
11320