Researches On Generalized Piecewise-Koszul Algebras

A kind of graded algebras, so called generalized piecewise-Koszul algebras, is intro-duced in this paper, which includesК2 algebras as special examples. The Yoneda-Ext algebras of generalized piecewise-Koszul algebras are generated by elements of two homological degrees, namely,1 and p. Generalized piecewise-Koszul algebras coincide withК2 algebras while taking p=2. If p≥3, then generalized piecewise-Koszul alge-bras become quadratic algebras. The difference between generalized piecewise-Koszul algebras and piecewise-Koszul algebras is that the trivial module of piecewise-Koszul algebras has pure minimal projective resolution, which means that every projective module in the minimal projective resolution is generated by elements of one degree; while generalized piecewise-Koszul algebras may not have this character.The homological properties and A∞-structures of generalized piecewise-Koszul al-gebras are discussed. PBW-deformation is also constructed. We proved that the quotient algebras and twist algebras of generalized piecewise-Koszul algebras keep the general-ized piecewise-Koszulity. Homomorphisms between algebras are used to study the gen-eralized piecewise-Koszul algebras. The generalized piecewise-Koszulity of noetherian semiperfect algebras is discussed and the so-called quasi generalized piecewise-Koszul algebras are defined. The equivalent descriptions are given by the property of Yoneda-Ext algebras.