(p, λ)-Koszul algebras and modules

Abstract

In this paper, the notions of (p, λ)-Koszul algebra and (p, λ)-Koszul module are introduced. Some criteria theorems for a positively graded algebra A to be (p, λ)-Koszul are given. The notion of weakly (p, λ)-Koszul module is defined as well and let WK ^p _λ (A) denote the category of weakly (p, λ)-Koszul modules. We show that M ∈ WK ^p _λ (A) if and only if it can be approximated by (p, λ)-Koszul submodules, which is equivalent to that G(M) is a (p, λ)-Koszul module, where G(M) denotes the associated graded module of M. As applications, the relationships of the minimal graded projective resolutions of M, G(M) and (p, λ)-Koszul submodules are established. In particular, for a module M ∈ WK ^p _λ (A) we prove that ⊕_i≥0 Ext ⁱ _A (M,A ₀) ∈ gr ₀(E(A)), we also get as a consequence that the finitistic dimension conjecture is valid in WK ^p _λ (A) under certain conditions.