On modules with d-Koszul-type submodules

Abstract

The so-called weakly d-Koszul-type module is introduced and it turns out that each weakly d-Koszul-type module contains a d-Koszul-type submodule. It is proved that, M ∈ Open image in new window ℐd(A) if and only if M admits a filtration of submodules: 0 ⊂ U0 ⊂ U1 ⊂ ... ⊂ Up = M such that all Ui/Ui−1 are d-Koszul-type modules, from which we obtain that the finitistic dimension conjecture holds in Open image in new window ℐd(A) in a special case. Let M ∈ Open image in new window ∐d(A). It is proved that the Koszul dual ℰ(M) is Noetherian, Hopfian, of finite dimension in special cases, and ℰ(M) ∈ gr0(E(A)). In particular, we show that M ∈ Open image in new window ℐd(A) if and only if ℰ (G(M)) ∈ gr0(E(A)), where G is the associated graded functor.