δ-KOSZULITY OF FINITELY GENERATED GRADED MODULES

The delta-Koszulity of finitely generated graded modules is discussed and the notion of weakly delta-Koszul module is introduced. Let M is an element of gr(A) and {Sd(1), Sd(2), ... , Sd(m)} denote the set of minimal homogeneous generating spaces of M where S-di consists of homogeneous elements of M of degree d(i). Put M-1 = < Sd(1)>, M-2 = < Sd(1), Sd(2)>, ... , M-m = < Sd(1), Sd(2), ... , S-dm >. Then M admits a chain of graded submodules: 0 = M-0 subset of M-1 subset of M-2 subset of ... subset of M-m = M. Moreover, it is proved that M is a weakly delta-Koszul module if and only if all M-i/Mi-1[-d(i)] are delta-Koszul modules, if and only if the associated graded module G(M) is a delta-Koszul module. Further, as applications, the relationships of minimal graded projective resolutions among M, G(M) and these quotients M-i/Mi-1 are established. The Ext module circle plus(i >= 0) Ext(A)(i) (M, A(0)) of a weakly delta-Koszul module M is proved to be finitely generated in degree zero.