I have that $S=k[x_1, \dots, x_n]$, $I$ is a lattice ideal, and $J$ is a monomial ideal. I am interested in the resolution of $S/(I+J)\cong S/I\otimes S/J$. In particular, I am interested in knowing under what conditions the (minimal, free) resolutions of $S/I$ and $S/J$ exist as subcomplexes of the resolution of $S/(I+J)$.
One condition that I know must exist is that for every monomial generator of $J$, that generator cannot divide either half of any of the binomial generators. If this condition were not satisfied, then the ideal $I+J$ would not be minimally generated by the union of the generators. So far, all of my computed examples (in Singular) have shown the claim to hold under the given condition, so I tend to believe it.
I have proven that it is true in the first (or second, depending on how you look at it) syzygy module, but the proof is ugly, so I was hoping that I could use something like this question/answer.