arXiv subjects

A. M. Bouhada

Publications and source records attributed to A. M. Bouhada.

3 recordsLinked to original sources

arXiv2026

Koszul duality for finite-dimensional absolutely Koszul algebras

Let \(Λ\) be a finite-dimensional Koszul algebra. Using a description of the linear part of minimal projective resolutions in terms of Koszul duality, we prove that \(Λ\) is absolutely Koszul if and only if its Koszul dual \(Λ^{!}\) is graded left co-coherent. We further show that an absolutely Koszul algebra of finite global linearity defect has finite graded finitistic dimension. Finally, we refine Koszul duality for finite-dimensional absolutely Koszul algebras. As a further application, we answer a question of Green et al.~\cite{12} by means of the Koszul dual algebra.

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arXiv2026

Koszul Duality for Coherent Sheaves

We establish a bounded derived Koszul duality for infinite-dimensional Koszul algebras and derive the corresponding singular Koszul duality. We then specialize this framework to two classes of Koszul algebras, namely quadratic monomial algebras and absolutely Koszul algebras satisfying an additional homological condition, for which the resulting dualities admit particularly well-behaved forms. As an application to algebraic geometry, let \(Λ\) be a commutative noetherian Koszul algebra generated in degree \(1\), and set \(X=\operatorname{Proj}(Λ)\). We obtain a Koszul-dual description of \(\mathsf{D}^{b}\!\bigl(\operatorname{coh}(X)\bigr)\), yielding a BGG-type correspondence for projective schemes defined by such algebras. As a second application, in noncommutative projective geometry, we consider generalized Artin--Schelter regular Koszul algebras \(Λ^{!}\) arising as Koszul duals of finite-dimensional self-injective Koszul algebras \(Λ\). We show that \(\mathsf{D}^{b}\!\bigl(\operatorname{qgr}(Λ^{!})\bigr)\) is triangulated equivalent to the bounded derived category of finite-dimensional modules over a finite-dimensional Koszul algebra of finite global dimension. This yields a Beilinson-type description of \(\mathsf{D}^{b}\!\bigl(\operatorname{qgr}(Λ^{!})\bigr)\), extending the classical description of coherent sheaves on projective space to this noncommutative setting.

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arXiv2026

A Non-graded Koszul Duality and Its Applications

Let \(Λ\) be a finite-dimensional Koszul algebra with Koszul dual \(Λ^!\). We establish derived Koszul dualities at the level of bounded derived categories, both in the graded setting \(\mathsf{D}^{b}(Λ\textup{-gmod})\) and in the ungraded setting \(\mathsf{D}^{b}(Λ\textup{-mod})\), without imposing finiteness conditions on \(Λ^!\). We first prove a graded derived Koszul duality for every finite-dimensional Koszul algebra, with no Noetherian or coherence assumptions on the Koszul dual. We then show that the bounded derived category \(\mathsf{D}^{b}(Λ\textup{-mod})\) can be reconstructed from the graded theory as the triangulated hull of an orbit category. This yields a genuinely non-graded derived Koszul duality. We further establish singular and dg refinements of these dualities. For Iwanaga--Gorenstein Koszul algebras, this gives a stable Koszul duality for graded Gorenstein-projective modules and their ungraded counterparts, providing a non-graded form of the Bernstein--Gel'fand--Gel'fand correspondence. As applications, we obtain new descriptions of the bounded derived categories \(\mathsf{D}^{b}(\mathcal{O}_λ)\) for all integral blocks of category \(\mathcal{O}\), including singular blocks, thereby closing a gap left open in the work of Beilinson, Ginzburg, and Soergel. We also establish analogous dualities for certain categories of perverse sheaves arising in geometric representation theory. Finally, we formulate conjectural descriptions of bounded derived and singularity categories of finite-dimensional graded algebras in terms of dg orbit categories.

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