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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Vanishing of Ext and Tor over fiber products
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by Saeed Nasseh and Sean Sather-Wagstaff PDF
Proc. Amer. Math. Soc. 145 (2017), 4661-4674

Abstract:

Consider a non-trivial fiber product $R=S\times _kT$ of local rings $S$, $T$ with common residue field $k$. Given two finitely generated $R$-modules $M$ and $N$, we show that if $\operatorname {Tor}^R_i(M,N)=0=\operatorname {Tor}^R_{i+1}(M,N)$ for some $i\geqslant 5$, then $\operatorname {pd}_R(M)\leqslant 1$ or $\operatorname {pd}_R(N)\leqslant 1$. From this, we deduce several consequences, for instance, that $R$ satisfies the Auslander-Reiten Conjecture.
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Additional Information
  • Saeed Nasseh
  • Affiliation: Department of Mathematical Sciences, Georgia Southern University, Statesboro, Georgia 30460
  • MR Author ID: 867753
  • Email: snasseh@georgiasouthern.edu
  • Sean Sather-Wagstaff
  • Affiliation: Department of Mathematical Sciences, Clemson University, O-110 Martin Hall, Box 340975, Clemson, South Carolina 29634
  • Email: ssather@clemson.edu
  • Received by editor(s): March 25, 2016
  • Received by editor(s) in revised form: April 20, 2016, and December 6, 2016
  • Published electronically: June 22, 2017
  • Additional Notes: The second author was supported in part by North Dakota EPSCoR, National Science Foundation Grant EPS-0814442, and NSA Grant H98230-13-1-0215.
  • Communicated by: Irena Peeva
  • © Copyright 2017 Copyright is retained by the authors.
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 4661-4674
  • MSC (2010): Primary 13D02, 13D05, 13D07, 13D09
  • DOI: https://doi.org/10.1090/proc/13633
  • MathSciNet review: 3691985