  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.

Commutative Algebra;Combinatorics