  Grothendieck proved that any locally free sheaf on a projective line over a
field (uniquely) decomposes into a direct sum of line bundles. Ishii and Uehara
construct an analogue of Grothendieck's theorem for pure sheaves on the
fundamental cycle of the Kleinian singularity $A_n$. We first study the
analogue for the other Kleinian singularities except for $A_n$. We also study
the classification of rigid pure sheaves on the reduced scheme of the
fundamental cycles. The classification is related to the classification of
spherical objects in a certain Calabi-Yau $2$-dimensional category.
