  Given a smooth projective toric variety $X$ of Picard rank 2, we resolve the
diagonal sheaf on $X \times X$ by a linear complex of length $\dim{X}$
consisting of finite direct sums of line bundles. As applications, we prove a
new case of a conjecture of Berkesch-Erman-Smith that predicts a version of
Hilbert's Syzygy Theorem for virtual resolutions, and we obtain a Horrocks-type
splitting criterion for vector bundles over smooth projective toric varieties
of Picard rank 2, extending a result of Eisenbud-Erman-Schreyer. We also apply
our results to give a new proof, in the case of smooth projective toric
varieties of Picard rank 2, of a conjecture of Orlov concerning the Rouquier
dimension of derived categories.
