  Let $ R $ be a $ d $-dimensional Cohen-Macaulay (CM) local ring of minimal
multiplicity. Set $ S := R/({\bf f}) $, where $ {\bf f} := f_1,\ldots,f_c $ is
an $ R $-regular sequence. Suppose $ M $ and $ N $ are maximal CM $ S
$-modules. It is shown that if $ \mathrm{Ext}_S^i(M,N) = 0 $ for some $ (d+c+1)
$ consecutive values of $ i \geqslant 2 $, then $ \mathrm{Ext}_S^i(M,N) = 0 $
for all $ i \geqslant 1 $. Moreover, if this holds true, then either $
\mathrm{projdim}_R(M) $ or $ \mathrm{injdim}_R(N) $ is finite. In addition, a
counterpart of this result for Tor-modules is provided. Furthermore, we give a
number of necessary and sufficient conditions for a CM local ring of minimal
multiplicity to be regular or Gorenstein. These conditions are based on
vanishing of certain Exts or Tors involving homomorphic images of syzygy
modules of the residue field.
