About: Kaplansky classes, finite character and aleph(1)-projectivity

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rdf:type	skos:Concept http://linked.opendata.cz/ontology/domain/vavai/Vysledek
rdfs:seeAlso	http://dx.doi.org/10.1515/FORM.2011.101
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. (en)
Title	Kaplansky classes, finite character and aleph(1)-projectivity Kaplansky classes, finite character and aleph(1)-projectivity (en)
skos:prefLabel	Kaplansky classes, finite character and aleph(1)-projectivity Kaplansky classes, finite character and aleph(1)-projectivity (en)
skos:notation	RIV/00216208:11320/12:10128138!RIV13-GA0-11320___
http://linked.open...avai/riv/aktivita	P Z
http://linked.open...avai/riv/aktivity	P(GA201/09/0816), P(GD201/09/H012), P(MEB101005), Z(MSM0021620839)
http://linked.open...iv/cisloPeriodika	5
http://linked.open...vai/riv/dodaniDat	2013
http://linked.open...aciTvurceVysledku	Trlifaj, Jan Šaroch, Jan
http://linked.open.../riv/druhVysledku	J - Článek v odborném periodiku
http://linked.open...iv/duvernostUdaju	S - Úplné a pravdivé údaje nepodléhající ochraně podle zvláštních právních předpisů
http://linked.open...titaPredkladatele	Univerzita Karlova v Praze / Matematicko-fyzikální fakulta
http://linked.open...dnocenehoVysledku	144348
http://linked.open...ai/riv/idVysledku	RIV/00216208:11320/12:10128138
http://linked.open...riv/jazykVysledku	eng - angličtina
http://linked.open.../riv/klicovaSlova	aleph1-projectivity; character; finite; classes; Kaplansky (en)
http://linked.open.../riv/klicoveSlovo	character finite Kaplansky aleph1-projectivity classes
http://linked.open...odStatuVydavatele	DE - Spolková republika Německo
http://linked.open...ontrolniKodProRIV	[DB92A95EE8E5]
http://linked.open...i/riv/nazevZdroje	Forum Mathematicum
http://linked.open...in/vavai/riv/obor	BA
http://linked.open...ichTvurcuVysledku	2 (xsd:int)
http://linked.open...cetTvurcuVysledku	2 (xsd:int)
http://linked.open...vavai/riv/projekt	Structure of Modules Algebraic and geometric methods and structures Algebraické metody teorie reprezentací (aproximace, realizace a omezení)
http://linked.open...UplatneniVysledku	2012
http://linked.open...v/svazekPeriodika	24
http://linked.open...iv/tvurceVysledku	Trlifaj, Jan Šaroch, Jan
http://linked.open...ain/vavai/riv/wos	000309161800008
http://linked.open...n/vavai/riv/zamer	Metody moderní matematiky a jejich aplikace
issn	0933-7741
number of pages	19 (xsd:int)
http://bibframe.org/vocab/doi	10.1515/FORM.2011.101
http://localhost/t...ganizacniJednotka	11320

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