  There are many structures (algebras, categories, etc) with natural gradings
such that the degree 0 components are not semisimple. Particular examples
include tensor algebras with non-semisimple degree 0 parts, extension algebras
of standard modules of standardly stratified algebras. In this thesis we
develop a generalized Koszul theory for graded algebras (categories) whose
degree 0 parts may be non-semisimple. Under some extra assumption, we show that
this generalized Koszul theory preserves many classical results such as the
Koszul duality. Moreover, it has some close relation to the classical theory.
Applications of this generalized theory to finite EI categories, directed
categories, and extension algebras of standard modules of standardly stratified
algebras are described. We also study the stratification property of standardly
stratified algebras, and classify algebras standardly (resp., properly)
stratified for all linear orders.
