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Proceedings of the American Mathematical Society

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Vanishing of Ext and Tor over fiber products


Authors: Saeed Nasseh and Sean Sather-Wagstaff
Journal: Proc. Amer. Math. Soc. 145 (2017), 4661-4674
MSC (2010): Primary 13D02, 13D05, 13D07, 13D09
DOI: https://doi.org/10.1090/proc/13633
Published electronically: June 22, 2017
MathSciNet review: 3691985
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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

Keywords: Auslander-Reiten, Ext-index, Ext-vanishing, fiber product, injective dimension, projective dimension, semidualizing complexes, Tor-vanishing
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
Article copyright: © Copyright 2017 Copyright is retained by the authors.