AI 中文总结
研究为无限维 Koszul 代数建立 Koszul 对偶性，应用于二次单项式代数等两类代数，还应用于代数几何，得到交换诺特 Koszul 代数定义的射影概型的有界导出范畴的 Koszul 对偶描述及 BGG 型对应。
AI 中文摘要
我们为无限维 Koszul 代数建立了有界导出 Koszul 对偶性，并得到了相应的奇异 Koszul 对偶性。然后将此框架应用于两类 Koszul 代数，即二次单项式代数和满足附加同调条件的绝对 Koszul 代数。对于这些类，一般对偶性特化为行为良好的形式。作为代数几何的应用，设\(\Lambda\)是在 1 次生成的交换诺特 Koszul 代数，\(X = \operatorname{Proj}(\Lambda)\)，我们得到了有界导出范畴\(\mathsf{D}^{b}\!\bigl(\operatorname{coh}(X)\bigr)\)的 Koszul 对偶描述。这给出了由交换诺特 Koszul 代数定义的射影概型的 BGG 型对应。
英文摘要
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 \(X\subseteq \mathbb{P}_k^n\) be an arbitrary closed projective subscheme. We obtain a Koszul-dual description of the bounded derived category \(\mathsf{D}^{b}\!\bigl(\operatorname{coh}(X)\bigr)\), thereby yielding a BGG-type correspondence for arbitrary closed projective subschemes in projective space. 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.
Comments36 pages. Version 5 (identical to Version 4 and final): We added further references and revised the abstract. In particular, by combining Serre's theorem with Backelin's result on the Koszulity of Veronese subalgebras, we extended the geometric application from closed projective subschemes arising from Koszul algebras to all closed projective subschemes of projective space
