A construction that repairs King's Conjecture - the (false) claim that every smooth projective toric variety has a full strong exceptional collection of line bundles - by working with a whole Cox ring at once rather than a single toric variety. The Cox category DCox(X) glues the derived categories of all toric varieties arising from a fixed Cox ring, indexed by the maximal chambers of the secondary fan and stitched together by Fourier-Mukai transforms from the birational maps between them. Learners see why the Bondal-Thomsen collection (the natural generalization of Beilinson's line bundles on P^n) becomes a tilting object, and a full strong exceptional collection in the projective case, so that King's Conjecture and Bondal's proposal hold without exception once all the toric geometry of the Cox ring is synthesized.
Introduces the Cox category DCox(X), a triangulated category glued from the bounded derived categories of all the toric varieties (toric stacks) that share a given Cox ring, indexed by the maximal ch…