Polycubes and polycube ideals

To a polycube $\mathcal{P}$ in ${\mathbb{R}}^n$, we associate an ideal $I_{\mathcal{P}}$ generated by the join–meet relations of vertices that are corners of a cuboid of $\mathcal{P}$. For $n=2$ these are the polyomino ideals. In contrast with the planar case, we show that for $n\ge 3$ the ideal $I_{\mathcal{P}}$ is prime if and only if $\mathcal{P}$ is a cuboid. For convex polycubes in ${\mathbb{R}}^3$ of uniform thickness whose vertex set is a sublattice of ${\mathbb{N}}^3$, the defining relations form a quadratic Gröbner basis, and we characterize the Cohen–Macaulay ones in terms of a poset on the vertices of $\mathcal{P}$; in this case the order complex of the poset is vertex decomposable. Finally, as an application, we show that the polycube algebra of a thickened $L$-convex polyomino is Koszul, and we characterize when it is Cohen–Macaulay.