Stratifying systems and proper systems

MARÍA INÉS PLATZECK

Woodshole, Mass, USA

Conferencia; Maurice Auslander Distinguished Lectures and International Conference; 2010

Northeastern University. Mathematics Department

Let Let ƒ© be a finite dimensional algebra over an algebraically closed field k. For any fixed ordering of the simple A-modules S1, . . . , Sn, the standard modules ƒ¢i are defined as the largest quotient of the projective cover of Si having composition factors in {S1, . . . Si−1}, for i ! n. An algebra is standardly stratified if ƒ© Let Let ƒ© be a finite dimensional algebra over an algebraically closed field k. For any fixed ordering of the simple A-modules S1, . . . , Sn, the standard modules ƒ¢i are defined as the largest quotient of the projective cover of Si having composition factors in {S1, . . . Si−1}, for i ! n. An algebra is standardly stratified if ƒ©A-modules S1, . . . , Sn, the standard modules ƒ¢i are defined as the largest quotient of the projective cover of Si having composition factors in {S1, . . . Si−1}, for i ! n. An algebra is standardly stratified if ƒ©Si having composition factors in {S1, . . . Si−1}, for i ! n. An algebra is standardly stratified if ƒ© has a filtration whose quotients are isomorphic to standard modules. The notion of standardly stratified system was defined by Erdmann and SLaenz, generalizing the standard modules, and the category of modules having a filtration with factors in the system has very interesting properties, as in the case of standardly stratified algebras. Another family of modules playing an interesting role for standardly stratified algebras are the proper standard modules, defined as certain factors of the standard modules. Here we define the notion of proper system, generalizing the proper standard modules, study the category of modules filtered by such a system, and associate to it a standardly stratified algebra. 1