The Eilenberg-Watts theorem over schemes

TL;DR

This paper extends the Eilenberg-Watts theorem to schemes, identifying conditions under which functors between quasi-coherent sheaves are tensor functors, with applications to noncommutative geometry.

Contribution

It generalizes the Eilenberg-Watts theorem to the setting of schemes and explores obstructions to functors being tensor products, especially in non-affine cases.

Findings

Obstructions vanish when the scheme is affine or functor is exact.

The result aids in classifying noncommutative $ ext{P}^1$-bundles.

Provides criteria for functors to be tensoring with bimodules.

Abstract

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 by C. Ingalls and D. Patrick are noncommutative $\mathbb{P}^{1}$-bundles in the sense of M. Van den Bergh.