  A notion of a split quasi-hereditary algebra has been defined by Cline,
Parshall and Scott. Du and Rui describe a based approach to split
quasi-hereditary algebras. We develop this approach further to show that over a
complete local Noetherian ring, one can achieve even stronger basis properties.
This is important for `schurifying' quasi-hereditary algebras as developed in
our subsequent work. The schurification procedure associates to an algebra $A$
a new algebra, which is the classical Schur algebra if $A$ is a field.
Schurification produces interesting new quasi-hereditary and cellular algebras.
It is important to work over an integral domain of characteristic zero, taking
into account a super-structure on the input algebra $A$. So we pay attention to
super-structures on quasi-hereditary algebras and investigate a subtle
conforming property of heredity data which is crucial to guarantee that the
schurification of $A$ is quasi-hereditary if so is $A$. We establish a Morita
equivalence result which allows us to pass to basic quasi-hereditary algebras
preserving conformity.
