On a Generalized Auslander-Reiten Conjecture
Identifikátory výsledku
Kód výsledku v IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F24%3A10489704" target="_blank" >RIV/00216208:11320/24:10489704 - isvavai.cz</a>
Výsledek na webu
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=LhwVavY0dk" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=LhwVavY0dk</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s10468-024-10271-z" target="_blank" >10.1007/s10468-024-10271-z</a>
Alternativní jazyky
Jazyk výsledku
angličtina
Název v původním jazyce
On a Generalized Auslander-Reiten Conjecture
Popis výsledku v původním jazyce
It is well-known that the generalized Auslander-Reiten condition (GARC) and the symmetric Auslander condition (SAC) are equivalent, and (GARC) implies that the Auslander-Reiten condition (ARC). In this paper we explore (SAC) along with the several canonical change of rings R -> Sdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$R rightarrow S$$end{document}. First, we prove the equivalence of (SAC) for R and R/xR, where x is a non-zerodivisor on R, and the equivalence of (SAC) and (SACC) for rings with positive depth, where (SACC) is the symmetric Auslander condition for modules with constant rank. The latter assertion affirmatively answers a question posed by Celikbas and Takahashi. Secondly, for a ring homomorphism R -> Sdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$R rightarrow S$$end{document}, we prove that if S satisfies (SAC) (resp. (ARC)), then R also satisfies (SAC) (resp. (ARC)) if the flat dimension of S over R is finite. We also prove that (SAC) holds for R implies that (SAC) holds for S when R is Gorenstein and S=R/Q & ell;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$S=R/Q<^>ell $$end{document}, where Q is generated by a regular sequence of R and the length of the sequence is at least & ell;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ell $$end{document}. This is a consequence of more general results about Ulrich ideals proved in this paper. Applying these results to determinantal rings and numerical semigroup rings, we provide new classes of rings satisfying (SAC). A relation between (SAC) and an invariant related to the finitistic extension degree is also explored.
Název v anglickém jazyce
On a Generalized Auslander-Reiten Conjecture
Popis výsledku anglicky
It is well-known that the generalized Auslander-Reiten condition (GARC) and the symmetric Auslander condition (SAC) are equivalent, and (GARC) implies that the Auslander-Reiten condition (ARC). In this paper we explore (SAC) along with the several canonical change of rings R -> Sdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$R rightarrow S$$end{document}. First, we prove the equivalence of (SAC) for R and R/xR, where x is a non-zerodivisor on R, and the equivalence of (SAC) and (SACC) for rings with positive depth, where (SACC) is the symmetric Auslander condition for modules with constant rank. The latter assertion affirmatively answers a question posed by Celikbas and Takahashi. Secondly, for a ring homomorphism R -> Sdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$R rightarrow S$$end{document}, we prove that if S satisfies (SAC) (resp. (ARC)), then R also satisfies (SAC) (resp. (ARC)) if the flat dimension of S over R is finite. We also prove that (SAC) holds for R implies that (SAC) holds for S when R is Gorenstein and S=R/Q & ell;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$S=R/Q<^>ell $$end{document}, where Q is generated by a regular sequence of R and the length of the sequence is at least & ell;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ell $$end{document}. This is a consequence of more general results about Ulrich ideals proved in this paper. Applying these results to determinantal rings and numerical semigroup rings, we provide new classes of rings satisfying (SAC). A relation between (SAC) and an invariant related to the finitistic extension degree is also explored.
Klasifikace
Druh
J<sub>imp</sub> - Článek v periodiku v databázi Web of Science
CEP obor
—
OECD FORD obor
10101 - Pure mathematics
Návaznosti výsledku
Projekt
<a href="/cs/project/GA23-05148S" target="_blank" >GA23-05148S: Homologická a strukturní teorie v geometrických kontextech</a><br>
Návaznosti
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Ostatní
Rok uplatnění
2024
Kód důvěrnosti údajů
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Údaje specifické pro druh výsledku
Název periodika
Algebras and Representation Theory
ISSN
1386-923X
e-ISSN
1572-9079
Svazek periodika
27
Číslo periodika v rámci svazku
3
Stát vydavatele periodika
NL - Nizozemsko
Počet stran výsledku
22
Strana od-do
1581-1602
Kód UT WoS článku
001226636500001
EID výsledku v databázi Scopus
2-s2.0-85193233594