We study obstructions to a direct limit preserving right exact functor F between categories of quasi-coherent sheaves on schemes being isomorphic to tensoring with a bimodule. When the domain scheme is affine, or if F is exact, all obstructions vanish and we recover the Eilenberg-Watts Theorem. This result is crucial to the proof that the noncommutative Hirzebruch surfaces constructed in C. Ingalls, D. Patrick (2002)  are noncommutative P1-bundles in the sense of M. Van den Bergh . © 2010 Elsevier B.V.
Nyman, A. (2010). The Eilenberg-Watts theorem over schemes. Journal of Pure and Applied Algebra, 214(11), 1922–1954. https://doi.org/10.1016/j.jpaa.2009.12.030