Abstract
Consider a non-trivial fiber product R = S × k T R=S\times _kT of local rings S S , T T with common residue field k k . Given two finitely generated R R -modules M M and N N , we show that if Tor i R ( M , N ) = 0 = Tor i + 1 R ( M , N ) \operatorname {Tor}^R_i(M,N)=0=\operatorname {Tor}^R_{i+1}(M,N) for some i ⩾ 5 i\geqslant 5 , then pd R ( M ) ⩽ 1 \operatorname {pd}_R(M)\leqslant 1 or pd R ( N ) ⩽ 1 \operatorname {pd}_R(N)\leqslant 1 . From this, we deduce several consequences, for instance, that R R satisfies the Auslander-Reiten Conjecture.
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