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 Jorgensen, we define the Castelnuovo-Mumford regularity reg(X) of a complex X \in D^b(grA) in terms of the local cohomologies or the minimal projective resolution of X. Let A^! be the quadratic dual ring of A. For the Koszul duality functor DG : D^b(grA) -> D^b(grA^!), we have reg(X) = max {i | H^i(DG (X)) \ne 0}. Using these concepts, we study weakly Koszul modules (= componentwise linear modules) over A^!. As an application, refining a result of Herzog and Roemer, we show that if J is a monomial ideal of an exterior algebra E= \bigwedge < y_1, ..., y_d > with d \geq 3, then the (d-2)nd syzygy of E/J is weakly Koszul. 
Kohji Yanagawa