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 Castelnuovo-Mumford regularity reg(M
- ) of a complex M
- ∈Db(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:Db(grA)→Db(grA!), we have reg(M
- )=max{i∣Hi(G(M
- ))≠0}. Using these concepts, we interpret results of Martinez-Villa and Zacharia concerning weakly Koszul modules (also called componentwise linear 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.