  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.
