Vanishing of Ext and Tor over fiber products

2016/03/17 by Saeed Nasseh, Nasseh, Saeed, Sean Sather-Wagstaff +1 · 2 citations
Mathematics · #13D02 #13D05 #13D07 #13D09 #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #msc:13D02 #msc:13D05 #msc:13D07 #msc:13D09

paper · pdf · doi:10.48550/arxiv.1603.05711

13 pages. comments welcome. v.2 includes section about depth formula and minor other revisions. v.3 includes corrections to Lemmas 2.3-2.5

arxiv created 2016/04/20 · arxiv updated 2016/04/22

Abstract

Consider a non-trivial fiber product R=S×_kT of local rings S, T with common residue field k. Given two finitely generate R-modules M and N, we show that if Tor^R_i(M,N)=0=Tor^R_i+1(M,N) for some i≥ 5, then pd_R(M)≤ 1 or pd_R(N)≤ 1. From this, we deduce several consequence, for instance, that R satisfies the Auslander-Reiten Conjecture.

Cited by

• Tor algebra of local rings with decomposable maximal ideal
• On Tor-vanishing of local rings

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• On Tor-vanishing of local rings
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• On modules with self Tor vanishing
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