The University of Montana
Department of Mathematical Sciences

Technical report #9/2007

Watts Therorem for Schemess

Adam Nyman
Department of Mathematical Sciences
University of Montana

S. Paul Smith
Department of Mathematics
University of Washington

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: Watts theorem, Morita theory for schemes, non-commutative Hirzebruch surface

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

Download Technical Report: Pdf (393 KB)