Graded, Selfinjective, and Koszul Algebras,

In this paper we continue our work on Koszul algebras initiated in earlier studies. The consideration about the existence of almost split sequences for Koszul modules appeared in our early work and only a partial answer is known. Koszul duality relates finite dimensional algebras of infinite global dimension with infinite dimensional algebras such that the injective dimension of the graded simples is finite. Two questions naturally arise: the existence of almost split sequences for infinite dimensional graded algebras and the relations between the global dimension of a graded algebra and the injective dimension of the graded simples. In the first part of the paper, we deal with the first matter; in the second, we consider the relations between the graded global dimension and the maximum of the injective dimensions of the graded simples, proving that in the noetherian case they coincide. In the third part, we consider finite dimensional graded selfinjective quiver algebras and prove that for such algebras all indecomposable projective modules have the same Lowey length. In the forth and last part of the paper, we specialize to selfinjective Koszul algebras and characterize their Yoneda algebras, proving they constitute non-commutative versions of the commutative regular algebras.