Derived categories of sheaves of quasi -projective schemes

We study perfect derived category and bounded derived category of coherent sheaves on quasi-projective schemes over a field k. For a quasi-projective scheme X over k, we show that Dbcoh,c (X) is equivalent to a particular category of functors on Dperf(X). If X is projective over a perfect field, then Dperf( X) is equivalent to a similar category of functors on, Dbcoh (X). This "perfect categorical pairing" allows us to define weak notions of adjunctions for functors on these categories and to prove their existence is most cases. We also relativize the notion of a Serre functor and study the properties of these functors.;With these ideas in hand, we extend Bondal and Orlov's reconstruction result to Goren-stein projective varieties and we extend Orlov's representability result for equivalences to the case where both schemes are just projective.