Koszul Duality for Quadratic Monomial Algebras

TL;DR

This paper explores Koszul duality for quadratic monomial algebras, establishing their structural properties, dualities, and derived equivalences with explicit descriptions, advancing understanding of their module categories.

Contribution

It provides new examples of Koszul dualities for quadratic monomial algebras, proves their coherence properties, and refines related derived and singular dualities with explicit constructions.

Findings

Quadratic monomial algebras are absolutely Koszul with global linearity defect at most one.

Finitely presented modules have rational Poincaré and Hilbert series.

Associated categories are abelian, hereditary, and admit explicit structural descriptions.

Abstract

This paper provides a new class of examples for the Koszul dualities established in~\cite{5}. We study quadratic monomial algebras from the perspective of Koszul duality, with particular emphasis on finitely presented and finitely copresented graded modules over the Koszul dual algebra. For a finite-dimensional quadratic monomial algebra \(\Lambda\), we prove that the Koszul dual \(\Lambda^{!}\) is both left coherent and left co-coherent, and that finitely presented (resp.\ finitely copresented) modules coincide with perfect (resp.\ coperfect) modules. As a consequence, the associated tails and cotails categories are abelian and hereditary, and admit explicit structural descriptions. We further show that quadratic monomial algebras are absolutely Koszul and have global linearity defect at most one. In particular, every finitely presented module has rational Poincar\'e and Hilbert series. Building on these results, we refine the graded derived and singular Koszul dualities, as well as the graded BGG correspondence, by giving explicit realizations of the associated triangulated equivalences. These equivalences induce nonstandard \(t\)-structures whose hearts admit concrete descriptions in terms of linear complexes and modules. We also obtain corresponding refinements in the ungraded setting.