Cohomology and finiteness conditions for generalisations of Koszul algebras

We study finite dimensional Koszul algebras and their generalisations including d-Koszul algebras and (D, A)-stacked algebras, together with their projective resolutions and Hochschild cohomology. Then we introduce the stretched algebra ~Λ and give a functorial construction of the projective resolution of ~Λ =~r and the projective bimodule resolution of A. Following this, we show that if E(Λ) is finitely generated then so is E(~Λ). We investigate the connection between HH*( Λ) and HH*(~ Λ) and the finiteness condition (Fg) using the theory of stratifying ideals. We give sufficient conditions for a finite dimensional Koszul monomial algebra to have (Fg) and generalize this result to finite dimensional d-Koszul monomial algebras. It is known that if Λ is a d-Koszul algebra then ~ Λ is a (D, A)-stacked algebra, where D = dA. We investigate the converse. We give the construction of the algebra B from a (D, A)-stacked algebra A and show that if A is a (D, A)-stacked monomial algebra, then B is d-Koszul with D = dA.