Hostname: page-component-5db58dd55d-jnbmb Total loading time: 0 Render date: 2026-06-08T12:58:08.342Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Generalized Auslander–Reiten Conjecture under Certain Ring Extensions

Published online by Cambridge University Press:  20 November 2018

Saeed Nasseh
Saeed Nasseh*
Affiliation:
Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460, USA. e-mail: snasseh@georgiasouthern.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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 $R\left[ x \right]$ .

Keywords

13D0716E3016E65Auslander–Reiten conjecturefinitistic extension degreeGorenstein rings

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 1 , 01 March 2015 , pp. 134 - 143
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Auslander, M., Ding, S., and Solberg, O., Liftings and weak liftings of modules. J. Algebra 156 (1993), no. 2, 273–317. http://dx.doi.org/10.1006/jabr.1993.1076 Google Scholar
[2] Auslander, M. and Reiten, I., On a generalized version of the Nakayama conjecture. Proc. Amer. Math. Soc. 52 (1975), 69–74. http://dx.doi.org/10.1090/S0002-9939-1975-0389977-6 Google Scholar
[3] Dao, H. and Veliche, O., Comparing complexities of pairs of modules. J. Algebra 322 (2009), no. 9, 3047–3062. http://dx.doi.org/10.1016/j.jalgebra.2008.08.011 Google Scholar
[4] Diveris, K., Finitistic extension degree. Algebr. Represent. Theory 17 (2014), no. 2, 495–506. http://dx.doi.org/10.1007/s10468-013-9406-2 Google Scholar
[5] Huneke, C. and Leuschke, G., On a conjecture of Auslander and Reiten. J. Algebra 275 (2004), no. 2, 781–790. http://dx.doi.org/10.1016/j.jalgebra.2003.07.018 Google Scholar
[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] Jorgensen, D. and Şega, L., Nonvanishing cohomology and classes of Gorenstein rings. Adv. Math. 188 (2004), no. 2, 470–490. http://dx.doi.org/10.1016/j.aim.2003.11.003 Google Scholar
[8] Matsumura, H., Commutative ring theory. Second ed., Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, Cambridge, 1989.Google Scholar
[9] Nasseh, S. and Yoshino, Y., On Ext-indices of ring extensions. J. Pure Appl. Algebra 213 (2009), no. 7, 1216–2223. http://dx.doi.org/10.1016/j.jpaa.2008.11.034 Google Scholar
[10] Wei, J., Auslander bounds and homological conjectures. Rev. Mat. Iberoam. 27 (2011), no. 3, 871–884. Google Scholar
[11] Wei, J., Generalized Auslander-Reiten conjecture and tilting equivalences. Proc. Amer. Math. Soc. 138 (2010), no. 5, 1581–1585. Google Scholar
[12] Yoshino, Y., On degenerations of modules. J. Algebra 278 (2004), no. 1, 217–226. http://dx.doi.org/10.1016/j.jalgebra.2003.10.020 Google Scholar