Watts Theorem for Schemes: Preliminary Version

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 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 on vector bundles, then it is isomorphic to tensoring with a bimodule. This result is used in [2] to prove that the noncommutative Hirzebruch surfaces constructed in [4] are noncommutative P 1-bundles in the sense of [7]. We conclude by giving necessary and sufficient conditions under which a direct-limit and coherence preserving right-exact functor from P1 to P0 is an extension of tensoring with a bimodule by a sum of cohomologies.