  We establish a genericity property in the representation theory of a flat
family of finite-dimensional algebras in the sense of Cline-Parshal-Scott. More
precisely, we show that the decomposition matrices as introduced by Geck and
Rouquier of an algebra which is free of finite dimension over a noetherian
integral domain and which splits over the fraction field of this ring are
generically trivial, i.e., they are trivial in an open neighborhood of the
generic point of the spectrum of the base ring. This generalizes a classical
result by Brauer in modular representation theory of finite groups. We
furthermore show that this is true precisely on an open set in case all fibers
of the algebra split. In this way we get a stratification of the base scheme
such that decomposition maps are trivial on each stratum. Moreover, this
defines a new discriminant of such algebras which generalizes Schur elements of
simple modules for symmetric split semisimple algebras. We provide some
extensions to the theory of decomposition maps allowing us to work without the
usual normality assumption on the base ring.
