On modules with d-Koszul-type submodules

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 ∈ ℐ ^d (A) if and only if M admits a filtration of submodules: 0 ⊂ U ₀ ⊂ U ₁ ⊂ ... ⊂ U _p = M such that all U _i /U _i−1 are d-Koszul-type modules, from which we obtain that the finitistic dimension conjecture holds in ℐ ^d (A) in a special case. Let M ∈ ∐ ^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 ∈ ℐ ^d (A) if and only if ℰ (G(M)) ∈ gr₀(E(A)), where G is the associated graded functor. Supported by National Natural Science Foundation of China (Grant No. 10571152)