Assume that $I\subset k[x_1,\ldots,x_n]$ and $J\subset k[y_1,\ldots,y_m]$ are monomial ideals in different rings, and the minimal free resolution of $S/I$ and $S/J$, say $F_\cdot$ and $G_\cdot$, are both linear. I believe that $F\otimes G$ is a minimal free resolution for $S/I+J$. Does anyone have any comment for the proof, or any reference?

The tensor product of two chain complexes $(A,d_1)$ and $(B,d_2)$, say $A\oplus B$, is formed by taking all products $A_i \otimes B_j$ and letting $(A \otimes B)_k$ be the direst some of $A_i\otimes B_j$ for $i+j=k$. The differential maps are defined as $\partial(a\otimes b) = d_1a \otimes b + (-1)^i a \otimes d_2b$ when $a\in A_i$. Then we have $\partial^2 = 0$.

So in order to prove the question we need to check the exactness of the complex, and how it resolves the resolution of $I+J$.