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_16 , Thomas reproved Theorem using a reinterpretation of Kuznetsov's original proof. He handles the case where @math need not be geometric, but @math is required to be a genuine variety (not noncommutative'') and he works with a special class of Lefschetz decompositions (the rectangular'' ones).