A structure theorem for P^1-Spec k-bimodules

TL;DR

This paper characterizes the structure of certain functors from quasi-coherent sheaves on the projective line over an algebraically closed field to vector spaces, using the Eilenberg-Watts theorem, and identifies which are integral transforms.

Contribution

It provides a structure theorem for P^1-Spec k-bimodules, extending the Eilenberg-Watts theorem to this geometric setting.

Findings

Characterization of k-linear right exact functors from quasi-coherent sheaves on P^1_k to vector spaces.

Identification of which functors are integral transforms.

Extension of the Eilenberg-Watts theorem to the setting of schemes.

Abstract

Let k be an algebraically closed field. Using the Eilenberg-Watts theorem over schemes, we determine the structure of k-linear right exact direct limit and coherence preserving functors from the category of quasi-coherent sheaves on P^1_k to the category of vector spaces over k. As a consequence, we characterize those functors which are integral transforms.