The Eilenberg-watts Theorem over Schemes

We describe obstructions to a direct limit preserving right exact functor between categories of quasi-coherent sheaves on schemes being isomorphic to tensoring with a bimodule. When the domain scheme is affine, all obstructions vanish and we recover the Eilenberg-Watts Theorem. We use our description of these obstructions to prove that if a direct limit preserving right exact functor F from a smooth curve is exact, then it is isomorphic to tensoring with a bimodule. This result is crucial to the proof that the noncommutative Hirzebruch surfaces constructed in [3] are noncommutative P-bundles in the sense of [6].