We generalize Kuznetsov's theory of homological projective duality to the setting of noncommutative algebraic geometry. Simultaneously, we develop the theory over general base schemes, and remove the usual smoothness, properness, and transversality hypotheses.
@cite_16: Beginning with the theorems of Beilinson and Orlov on derived categories, we show how these lead naturally to Kuznetsov's beautiful theory of Homological Projective Duality. We then survey some examples.
@cite_1: Homological Projective duality (HP-duality) theory, introduced by Kuznetsov [42], is one of the most powerful frameworks in the homological study of algebraic geometry. The main result (HP-duality theorem) of the theory gives complete descriptions of bounded derived categories of coherent sheaves of (dual) linear sections of HP-dual varieties. We show the theorem also holds for more general intersections beyond linear sections. More explicitly, for a given HP-dual pair @math , then analogue of HP-duality theorem holds for their intersections with another HP-dual pair @math , provided that they intersect properly. We also prove a relative version of our main result. Taking @math to be dual linear subspaces (resp. subbundles), our method provides a more direct proof of the original (relative) HP-duality theorem.
@cite_3: 
During the preparation of this paper, two other works on HPD appeared. @cite_1 , Jiang--Leung--Xie build on the argument from @cite_16 to prove a generalization of Theorem , where the the pair @math is replaced by any HPD pair. We independently discovered (a more general form of) this result, which appears as an application in @cite_3 .