A Resolution of the Diagonal for Smooth Toric Varieties

Abstract

Beilinson gave a resolution of the diagonal for complex projective spaces, which Bayer-Popescu-Sturmfels generalized to what they refer to as unimodular projective toric varieties. The unimodular condition in Bayer-Popescu-Sturmfels' convention is more restrictive than being smooth. In the smooth, non-unimodular setting, extra vertices appear in the finite cellular complex which Bayer-Popescu-Sturmfels previously used to resolve the diagonal for unimodular projective toric varieties. We use the floor function to assign monomial labelings in a convex manner, and show that this assignment is compatible with the graded algebra involving the irrelevant ideal to give a resolution of the diagonal for a smooth projective toric variety.