Counterexamples for Cohen-Macaulayness of lattice ideals
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Let $mathscr{L}subset mathbb{Z}^n$ be a lattice, $I$ its corresponding lattice ideal, and $J$ the toric ideal arising from the saturation of $mathscr{L}$. 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.