The University of Montana

Department of Mathematical Sciences

Technical report #9/2007

**Adam Nyman**

Department of Mathematical

**S. Paul Smith**

Department of

**Abstract**

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 to prove that the noncommutative Hirzebruch
surfaces constructed by Ingalls and Patrick are noncommutative P^1-bundles in the sense of Van den Bergh. We conclude by giving necessary and
sufficient conditions under which a direct-limit and coherence preserving
right-exact functor from P^1 to P^0 is an extension

of tensoring with a bimodule by a sum of cohomologies.

**Keywords:**

**AMS Subject Classification:** 18F99, 14A22, 16D90, 18A25.

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