Castelnuovo-mumford Regularity for Complexes and Weakly Koszul Modules

Let A be a noetherian AS regular Koszul quiver algebra (if A is commutative, it is essentially a polynomial ring), and grA the category of finitely generated graded left A-modules. Following Jørgensen, we define the CastelnuovoMumford regularity reg(M•) of a complex M• ∈ D(grA) in terms of the local cohomologies or the minimal projective resolution of M•. Let A be the quadratic dual ring of A. For the Koszul duality functor G : D(grA) → D(grA), we have reg(M•) = max{ i | H(G(M)) 6= 0 }. Using these concepts, we interpret results of Martinez-Villa and Zacharia concerning weakly Koszul modules over A. As an application, refining a result of Herzog and Römer, we show that if J is a monomial ideal of an exterior algebra E = ∧ 〈y1, . . . , yd〉, d ≥ 3, then the (d − 2) nd syzygy of E/J is weakly Koszul.