  We generalize a method by L. Ambrozio, A. Carlotto, and B. Sharp to study the
Morse index of closed f-minimal hypersurfaces isometrically immersed in a
general weighted manifold. The technique permits, in particular, to obtain a
linear lower bound on the Morse index via the first Betti number for closed
f-minimal hypersurfaces in products of some compact rank one symmetric spaces
with an Euclidean factor, endowed with the rigid shrinking gradient Ricci
soliton structure. These include, as particular cases, all cylindric shrinking
gradient Ricci solitons.
