Counterexamples for Cohen-Macaulayness of Lattice Ideals

Let $\mathcal{L}\subset {\mathbb{Z}}^n$ be a lattice, $I$ its corresponding lattice ideal, and $J$ the toric ideal arising from the saturation of $\mathcal{L}$. We produce infinitely many examples, in every codimension, of pairs $I,J$ where one of these ideals is Cohen–Macaulay but the other is not.