Let $k$ be a field of characteristic zero. Suppose that $A$ and $B$ are finitely generated $k$-algebras, $I$ is an ideal of $A$ and $J$ an ideal of $B$. Consider the ideal of $A\otimes_kB$ given by $$ L=I\otimes_k B+A\otimes_k J $$ For each $i\in\mathbb{Z}_{\geq0}$ we have a natural map $$ \bigoplus\limits_{0\leq j\leq i}\left(I^j/I^{j+1}\otimes_k B/J\right)\otimes_{(A\otimes B)/L}\left(A/I\otimes_kJ^{i-j}/J^{j-i+1}\right)\to L^i/L^{i+1}$$ given by multiplication on each summand. I am quite sure this should be an isomorphism, and I believe it is surjective, but I haven't been able to prove it is injective. Any ideas?
In the case I am considering $I$ and $J$ are nilpotent ideals, so if that helps then you may assume that. In fact in my case, $I$ and $J$ are the nilradicals of $A$ and $B$ respectively, and $A/I$ and $B/J$ are integral domains. You can use that if it's helpful.