Hello!

Let $R$ be a regular local ring, and let $I,J\subset R$ be ideals. I'd like to understand the "meaning" of the existence of a regular sequence $(x_1,...,x_n)$ in $R$ such that $I$ is generated by $x_1,...,x_k$ and $J$ is generated by $(x_{k+1},...,x_n)$ for some $1\leq k\leq n$.

For example, the existence of such a sequence implies that $\text{Tor}^R_k(R/I,R/J)=0$ for all $k>0$.

Is it possible to give an *equivalent* description of the above property in terms of the vanishing of certain Tor and/or Ext terms?

Thank you! Hanno