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Let $S$ be the polynomial ring in $n$ variables, and let $I_1$ and $I_2$ be ideals in $S$. What can be said about the $\mathbb{Z}$-graded minimal free resolution of $I_1+ I_2$ in terms of the $\mathbb{Z}$ graded minimal free resolutions of $I_1$ and $I_2$? Is it completely determined by these? If so, I guess this is a standard result, and I would much appreciate if someone could tell me where I can find it/read about it. If it is not determined in general, are there any circumstances where it is? E.g. when $I_1$ and $I_2$ are monomial?

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