Gröbner bases, triangulations, and Koszul algebras

In this chapter we investigate the algebraic data of toric ideals that correspond to regular subdivisions of the cones generated by affine monoids: monomials orders and initial ideals.Though the correspondence is not strong enough for unconditional implications in either direction, it yields powerful results in cases where the unimodularity of triangulations or the squarefreeness of the initial ideals can be established. for example, the polytopal algebras defined by lattice polygons turn out to be Koszul algebras under the obvious necessary conditions.