Let
(
S
,
m
,
k
)
(S,\mathfrak {m},k)
and
(
T
,
n
,
k
)
(T,\mathfrak {n},k)
be local rings, and let
R
R
denote their fiber product over their common residue field
k
k
. Inspired by work of Nasseh and Sather-Wagstaff, we explore consequences of the vanishing of
T
o
r
m
R
(
M
,
N
)
\mathrm {Tor}^R_m(M,N)
for various values of
m
m
, where
M
M
and
N
N
are finitely generated
R
R
-modules.