Linear Koszul duality. II: Coherent sheaves on perfect sheaves. (Q2793756)

The main aim of the present paper is to prove that there exists a linear Koszul duality in a very general setting which is compatible with morphisms of vector bundles and base changes. The main construction, presented in Section 2.2, is the following: If \((X,\mathcal{O}_X)\) is a scheme, and \(\mathcal{X}\) is a finite non-positive complex constructed with locally free \(\mathcal{O}_X\)-modules of finite rank then \(\mathcal{Y}\) denotes the complex \(\mathcal{X}^\vee[-1]\), and \(\mathcal{T}\) and \(\mathcal{S}\) are the graded symmetric algebras associated to \(\mathcal{X}\) and \(\mathcal{Y}\). Then there are some natural exact functors between some categories of dg-modules \(\mathcal{C}(\mathcal{T}\text{-Mod})\) and \(\mathcal{C}(\mathcal{S}\text{-Mod})\) such that the restrictions to the subcategories of bounded above dg-modules are exact, and they induce an equivalence between the corresponding derived categories (Theorem 2.6). These functors are used to construct a linear Koszul duality in Theorem 2.8 for the case \(X\) is a nice scheme with a dualizing complex. In the last part of the paper the authors study various compatibility properties associated to this linear Koszul duality.NEWLINENEWLINEFor Part I, see [the authors, Compos. Math. 146, No. 1, 233--258 (2010; Zbl 1275.14014)].