Koszul Duality for Quadratic Monomial Algebras

Abstract

Let \(\Lambda\) be a finite-dimensional quadratic monomial algebra and let \(\Lambda^{!}\) be its Koszul dual. We investigate the structure of graded modules over \(\Lambda^{!}\) and derive several consequences for Koszul duality. We prove that \(\Lambda^{!}\) is both graded coherent and graded co-coherent. Moreover, finitely presented and finitely copresented graded \(\Lambda^{!}\)-modules coincide with perfect and coperfect modules, respectively. As a consequence, the associated tails and cotails categories are hereditary abelian categories admitting explicit descriptions in terms of linear and colinear modules. We further show that every finite-dimensional quadratic monomial algebra is absolutely Koszul and has global linearity defect at most one. In particular, finitely presented graded modules have rational Poincar\'e and Hilbert series. Using these structural results, we refine graded and ungraded derived Koszul dualities, singular Koszul dualities, and the Bernstein--Gelfand--Gelfand correspondence. We obtain explicit descriptions of the associated triangulated categories and of the induced nonstandard \(t\)-structures. As an application, we derive new bounds on the finitistic dimension of quadratic monomial algebras in terms of finite paths in the Koszul dual algebra.