  This paper aims at developing a "local--global" approach for various types of
finite dimensional algebras, especially those related to Hecke algebras. The
eventual intention is to apply the methods and applications developed here to
the cross-characteristic representation theory of finite groups of Lie type.
The authors first review the notions of quasi-hereditary and stratified
algebras over a Noetherian commutative ring. They prove that many global
properties of these algebras hold if and only if they hold locally at every
prime ideal. When the commutative ring is sufficiently good, it is often
sufficient to check just the prime ideals of height at most one. These methods
are applied to construct certain generalized q-Schur algebras, proving they are
often quasi-hereditary (the "good" prime case) but always stratified. Finally,
these results are used to prove a triangular decomposition matrix theorem for
the modular representations of Hecke algebras at good primes. In the bad prime
case, the generalized q-Schur algebras are at least stratified, and a block
triangular analogue of the good prime case is proved, where the blocks
correspond to Kazhdan-Lusztig cells.
