Noncommutative Koszul filtrations

A standard associative graded algebra R over a field k is called Koszul if k admits a linear resolution as an R-module. A (right) R-module M is called Koszul if it admits a linear resolution too. Here we study a special class of Koszul algebras — roughly say, algebras having a lot of Koszul cyclic modules. Commutative algebras with similar properties (so-called algebras with Koszul filtrations) has been studied in several papers [CRV, CTV, Bl, Co1, Co2]. A cyclic right R-module M = R/J is Koszul if and only if its defining ideal J is Koszul and generated by linear forms. So, we may deal with degree-one generated Koszul ideals instead of cyclic Koszul modules. A chain 0 = I0 ⊂ I1 ⊂ . . . ⊂ In = R of right-sided degree-one generated ideals in an algebra A is called Koszul flag if every ideal Ij is a Koszul module. Every algebra R with Koszul flag is Koszul, and the Koszul algebras of the most important type, PBW-algebras, always contain Koszul flags (Theorem 1.1). In the case of commutative algebra R, a natural way to find a Koszul flag in the algebra (and so, to prove its Koszulness) is called Koszul filtration. This concept is introduced and studied in several papers of Conca and others [CRV, CTV, Co1]. They found that the most of quadratic algebras occuring in algebraic geometry (such as coordinate rings of canonical embeddings of general projective curves and of projective embeddings of abelian