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_23: We introduce a notion of homological projective duality for smooth algebraic varieties in dual projective spaces, a homological extension of the classical projective duality. If algebraic varieties X and Y in dual projective spaces are homologically projectively dual, then we prove that the orthogonal linear sections of X and Y admit semiorthogonal decompositions with an equivalent nontrivial component. In particular, it follows that triangulated categories of singularities of these sections are equivalent. We also investigate homological projective duality for projectivizations of vector bundles.
In his unpublished habilitation thesis @cite_23 , Kuznetsov developed a version of HPD which works for input categories @math that are realizable (compatibly with their @math -linear structure) as an admissible subcategory of the derived category of a variety. Due to my inadequate Russian I was not able to read @cite_23 , but its existence served as an inspiration for this paper.