Castelnuovo–Mumford regularity for complexes and weakly Koszul modules

Abstract

Let A be a noetherian AS-regular Koszul quiver algebra (if A is commutative, it is essentially a polynomial ring), and gr A the category of finitely generated graded left A -modules. Following Jørgensen, we define the Castelnuovo–Mumford regularity reg ( M • ) of a complex M • ∈ D b ( gr A ) 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 b ( gr A ) → D b ( gr A ! ) , we have reg ( M • ) = max { i ∣ H i ( 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 = ⋀ 〈 y 1 , … , y d 〉 , d ≥ 3 , then the ( d − 2 ) nd syzygy of E / J is weakly Koszul.