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arXiv · 1603.05711

Vanishing of Ext and Tor over fiber products

Saeed Nasseh·Sean Sather-Wagstaff
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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 generate $R$-modules $M$ and $N$, we show that if $\operatorname{Tor}^R_i(M,N)=0=\operatorname{Tor}^R_{i+1}(M,N)$ for some $i\geq 5$, then $\operatorname{pd}_R(M)\leq 1$ or $\operatorname{pd}_R(N)\leq 1$. From this, we deduce several consequence, for instance, that $R$ satisfies the Auslander-Reiten Conjecture.

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Saeed Nasseh, Sean Sather-Wagstaff. 2016-03-17. Vanishing of Ext and Tor over fiber products. https://arxiv.org/abs/1603.05711

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