δ-Koszulity of Finitely Generated Graded Modules

Abstract

The δ-Koszulity of finitely generated graded modules is discussed and the notion of weakly δ-Koszul module is introduced. Let M ∈ gr(A) and {S d 1 , S d 2 ,…, S d m } denote the set of minimal homogeneous generating spaces of M where S d i consists of homogeneous elements of M of degree d i . Put ℳ1 = ⟨ S d 1 ⟩, ℳ2 = ⟨ S d 1 , S d 2 ⟩,…, ℳ m = ⟨ S d 1 , S d 2 ,…, S d m ⟩. Then M admits a chain of graded submodules: 0 = ℳ0 ⊂ ℳ1 ⊂ ℳ2 ⊂ … ⊂ ℳ m = M. Moreover, it is proved that M is a weakly δ-Koszul module if and only if all ℳ i /ℳ i−1[−d i ] are δ-Koszul modules, if and only if the associated graded module G(M) is a δ-Koszul module. Further, as applications, the relationships of minimal graded projective resolutions among M, G(M) and these quotients ℳ i /ℳ i−1 are established. The Ext module of a weakly δ-Koszul module M is proved to be finitely generated in degree zero.