Vanishing of Ext and Tor over fiber products

الأصل · EN

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 Torʳᵢ(M,N)=0=Torʳᵢ₊₁(M,N) for some i≥ 5, then pdᵣ(M)≤ 1 or pdᵣ(N)≤ 1. From this, we deduce several consequence, for instance, that R satisfies the Auslander-Reiten Conjecture.