Piecewise-Koszul Algebras

A special class of homogeneous algebras are introduced in this paper:called piecewise-Koszul algebra,including Koszul and d-Koszul algebras as special examples.It is proved that piecewise-Koszul objects also have a lot of perfect homological properties similar to Koszul and d-Koszul algebras.We give equivalent descriptions of piecewise-Koszul algebras in terms of its Yoneda-Ext algebras.Moreover,the Aoo-structure on E(A) is discussed as well.Finally,we generalize piecewise-Koszul algebras to a more wider case and the so-called piecewise-Koszul-type algebras and(p,d,λ)-Koszul algebras are introduced.Some Artin-Schelter regular algebras of global dimension 5 are piecewise-Koszul-type algebras.In order to study the piecewise-Koszulity for finitely graded modules,the notions of weakly piecewise-Koszul module and block piecewise-Koszul mod-ule are introduced.It is proved that such modules can be approximated by piecewise-Koszul modules.Furthermore,the relations between weakly piecewise-Koszul mod-ules and classic piecewise-Koszul modules are investigated.The piecewise-Koszul prop-erty of a Noetherian semiperfect non-graded algebra R is discussed and the so-called quasi-piecewise-Koszul algebra is defined.With the help of the 'the second grading',the structure and the property of E(R) are studied,where R is a quasi-piecewise-Koszul al-gebra. We give two equivalent descriptions of quasi-piecewise-Koszul algebras in terms of E(R).