  We prove that cellular Noetherian algebras with finite global dimension are
split quasi-hereditary over a regular commutative Noetherian ring with finite
Krull dimension and their quasi-hereditary structure is unique, up to
equivalence. In the process, we establish that a split quasi-hereditary algebra
is semi-perfect if and only if the ground ring is a local commutative
Noetherian ring. We give a formula to determine the global dimension of a split
quasi-hereditary algebra over a commutative regular Noetherian ring (with
finite Krull dimension) in terms of the ground ring and finite-dimensional
split quasi-hereditary algebras. For the general case, we give upper bounds for
the finitistic dimension of split quasi-hereditary algebras over arbitrary
commutative Noetherian rings. We apply these results to Schur algebras over
regular Noetherian rings and to Schur algebras over quotients rings of the
integers.
