Geodesic
On the Generalized Auslander–Reiten Conjecture under Certain Ring Extensions
Canadian mathematical bulletin, Tome 58 (2015) no. 1, pp. 134-143

Voir la notice de l'article provenant de la source Cambridge University Press

We show that under some conditions a Gorenstein ring $R$ satisfies the Generalized Auslander–Reiten conjecture if and only if $R\left[ x \right]$ does. When $R$ is a local ring we prove the same result for some localizations of $R\left[ x \right]$ .
DOI : 10.4153/CMB-2014-052-9
Mots-clés : 13D07, 16E30, 16E65, Auslander–Reiten conjecture, finitistic extension degree, Gorenstein rings 
Nasseh, Saeed. On the Generalized Auslander–Reiten Conjecture under Certain Ring Extensions. Canadian mathematical bulletin, Tome 58 (2015) no. 1, pp. 134-143. doi: 10.4153/CMB-2014-052-9
@article{10_4153_CMB_2014_052_9,
     author = {Nasseh, Saeed},
     title = {On the {Generalized} {Auslander{\textendash}Reiten} {Conjecture} under {Certain} {Ring} {Extensions}},
     journal = {Canadian mathematical bulletin},
     pages = {134--143},
     year = {2015},
     volume = {58},
     number = {1},
     doi = {10.4153/CMB-2014-052-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2014-052-9/}
}
TY  - JOUR
AU  - Nasseh, Saeed
TI  - On the Generalized Auslander–Reiten Conjecture under Certain Ring Extensions
JO  - Canadian mathematical bulletin
PY  - 2015
SP  - 134
EP  - 143
VL  - 58
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2014-052-9/
DO  - 10.4153/CMB-2014-052-9
ID  - 10_4153_CMB_2014_052_9
ER  - 
%0 Journal Article
%A Nasseh, Saeed
%T On the Generalized Auslander–Reiten Conjecture under Certain Ring Extensions
%J Canadian mathematical bulletin
%D 2015
%P 134-143
%V 58
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2014-052-9/
%R 10.4153/CMB-2014-052-9
%F 10_4153_CMB_2014_052_9

[1] [1] Auslander, M., Ding, S., and Solberg, O., Liftings and weak liftings of modules. J. Algebra 156 (1993), no. 2, 273–317. Google Scholar | DOI

[2] [2] Auslander, M. and Reiten, I., On a generalized version of the Nakayama conjecture. Proc. Amer. Math. Soc. 52 (1975), 69–74. Google Scholar | DOI

[3] [3] Dao, H. and Veliche, O., Comparing complexities of pairs of modules. J. Algebra 322 (2009), no. 9, 3047–3062. Google Scholar | DOI

[4] [4] Diveris, K., Finitistic extension degree. Algebr. Represent. Theory 17 (2014), no. 2, 495–506. Google Scholar | DOI

[5] [5] Huneke, C. and Leuschke, G., On a conjecture of Auslander and Reiten. J. Algebra 275 (2004), no. 2, 781–790. Google Scholar | DOI

[6] [6] Huneke, C., Sega, L., and Vraciu, A., Vanishing of Ext and Tor over some Cohen-Macaulay local rings. Illinois J. Math. 48 (2004), no. 1, 295–317. Google Scholar

[7] [7] Jorgensen, D. and Şega, L., Nonvanishing cohomology and classes of Gorenstein rings. Adv. Math. 188 (2004), no. 2, 470–490. Google Scholar | DOI

[8] [8] Matsumura, H., Commutative ring theory. Second ed., Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, Cambridge, 1989. Google Scholar

[9] [9] Nasseh, S. and Yoshino, Y., On Ext-indices of ring extensions. J. Pure Appl. Algebra 213 (2009), no. 7, 1216–2223. Google Scholar | DOI

[10] [10] Wei, J., Auslander bounds and homological conjectures. Rev. Mat. Iberoam. 27 (2011), no. 3, 871–884. Google Scholar

[11] [11] Wei, J., Generalized Auslander-Reiten conjecture and tilting equivalences. Proc. Amer. Math. Soc. 138 (2010), no. 5, 1581–1585. Google Scholar

[12] [12] Yoshino, Y., On degenerations of modules. J. Algebra 278 (2004), no. 1, 217–226. Google Scholar | DOI

Cité par Sources :