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The bounded derived category of coherent sheaves gives an isomorphism invariant for smooth projective varieties which are Fano or general type. King conjectured in 1997 that any smooth complete toric variety has a strong, full exceptional collection of line bundles. King's conjecture was proven false, but determining the exact criteria for a smooth complete (weak-)Fano toric variety to have a strong full exceptional collection of line bundles remains open, as does finding a method to generate this SFEC of line bundles for varieties which admit such a collection. In this talk I will give a cellular resolution of the diagonal for smooth projective toric varieties which yields a SFEC of line bundles on unimodular toric surfaces, as well as for a smooth non-unimodular example in dimension 2. This is joint work with Gabriel Kerr.