Koszul Duality for Quadratic Monomial Algebras

Let $\mathrm{\Lambda }$ be a quadratic monomial algebra and let ${\mathrm{\Lambda }}^{!}$ denote its Koszul dual. We study finitely presented and finitely copresented graded modules over $\mathrm{\Lambda }$ and ${\mathrm{\Lambda }}^{!}$, and use this structure to obtain explicit forms of derived and singular Koszul duality. We prove that quadratic monomial algebras are both graded coherent and graded cocoherent. Applying this to the Koszul dual ${\mathrm{\Lambda }}^{!}$, which is again quadratic monomial, we show that finitely presented and finitely copresented graded ${\mathrm{\Lambda }}^{!}$-modules coincide with perfect and coperfect modules, respectively. It follows that the corresponding tails and cotails categories are hereditary abelian categories, and their objects are described in terms of linear and colinear modules. We also prove that quadratic monomial algebras are absolutely Koszul and have global linearity defect at most two. Consequently, finite-dimensional graded $\mathrm{\Lambda }$-modules have rational Poincar'e series, while finitely presented and finitely copresented graded ${\mathrm{\Lambda }}^{!}$-modules have rational Hilbert series. As applications, we refine graded and ungraded derived Koszul dualities, singular Koszul dualities, and Bernstein–Gelfand–Gelfand correspondences for quadratic monomial algebras. We further describe the induced nonstandard $t$-structures on bounded derived and singularity categories and derive bounds for the finitistic dimension of $\mathrm{\Lambda }$ in terms of finite paths in the Koszul dual algebra.