Here's a (probably) basic commutative algebra question: Suppose I have two regular local rings (A,I) and (B,J) of the same dimension along with a homomorphism $\phi:A \to B$. Suppose $I = (x_0,\ldots,x_{n-1})$ and $J = (y_0,\ldots,y_{n-1})$. What conditions do I need on $\phi$ and/or $B$ (as an $A$-module) to ensure that $x_i^{-1}A/(x_0,\ldots,x_{i-1}) \otimes_A B = y_i^{-1} B/(y_0,\ldots,y_{i-1})$?