  In this paper, we study homological dimensions of algebras linked by
recollements of derived module categories, and establish a series of new upper
bounds and relationships among their finitistic or global dimensions. This is
closely related to a longstanding conjecture, the finitistic dimension
conjecture, in representation theory and homological algebra. Further, we apply
our results to a series of situations of particular interest: exact contexts,
ring extensions, trivial extensions, pullbacks of rings, and algebras induced
from Auslander-Reiten sequences. In particular, we not only extend and amplify
Happel's reduction techniques for finitistic dimenson conjecture to more
general contexts, but also generalise some recent results in the literature.
