Cohen-macaulayness and Computation of Newton Graded Toric Rings

Let H ⊆ Z be a positive semigroup generated by A ⊆ H, and let K[H] be the associated semigroup ring over a field K. We investigate heredity of the Cohen–Macaulay property from K[H] to both its A -Newton graded ring and to its face rings. We show by example that neither one inherits in general the Cohen–Macaulay property. On the positive side we show that for every H there exist generating sets A for which the Newton graduation preserves Cohen–Macaulayness. This gives an elementary proof for an important vanishing result on A-hypergeometric Euler–Koszul homology. As a tool for our investigations we develop an algorithm to compute algorithmically the Newton filtration on a toric ring.