Cohen-Macaulay Complexes and Koszul Rings

A generalized Koszul theory and its application

“…We want to apply Koszul theory, which has been proved to be very useful in the representation theory of algebras, to study the Ext groups of representations of finite EI categories. Examples of such applications can be found in [21,23], where Koszul theory has been applied to incidence algebras of posets. In general this theory applies to graded algebras, and we do not assume that the degree 0 part of the algebra is semisimple, unlike the classical Koszul theory described in [4,9,10,17].…”

“…There do already exist several generalized Koszul theories where the degree 0 part A 0 of a graded algebra A is not required to be semisimple, see [11,15,16,23]. Each Koszul algebra A defined by Woodcock in [23] is supposed to satisfy that A is both a left projective A 0 -module and a right projective A 0 -module.…”

“…Each Koszul algebra A defined by Woodcock in [23] is supposed to satisfy that A is both a left projective A 0 -module and a right projective A 0 -module. This requirement is too strong for us.…”

A generalized Koszul theory and its relation to the classical theory

“…In [11,18,19,29] several generalized Koszul theories have been described, where the degree 0 part A 0 of a graded algebra A is not required to be semisimple. In [29], A is supposed to be both a left projective A 0 -module and a right projective A 0 -module.…”

“…Particular examples of such structures include tensor algebras generated by non-semisimple algebras A 0 and (A 0 , A 0 )-bimodules A 1 , extension algebras of finitely generated modules (among which we are most interested in extension algebras of standard modules of standardly stratified algebras [8,15]), graded modular skew group algebras, category algebras of finite EI categories [14,27,28], and certain graded k-linear categories. Therefore, it is reasonable to develop a generalized Koszul theory to study representations and homological properties of above structures.In [11,18,19,29] several generalized Koszul theories have been described, where the degree 0 part A 0 of a graded algebra A is not required to be semisimple. In [29], A is supposed to be both a left projective A 0 -module and a right projective A 0 -module.…”