Example

Let k be a field of characteristic $\neq 2$ and take for A the local ring $k[[X, Y]]/(X^2 + Y^2)$ with residue field k. Take $B = k[[X, T]]/(T^2 + 1)$ where $T = Y/X$. Distinguish two cases: if — 1 is the square of an element i of k, B has two maximal ideals generated respectively by {X, T + i} and {X, T − i}, they have residue field k, and one has $e_{m_A}^A(A) = e_{m_{AB}}^B(B) = 2$. If — 1 is not a square in k, B has a unique maximal ideal (X) with residue field $k[T]/(T^2 + 1)$, and one has $e_{m_A}^A(A) = 2,\ e_{m_{AB}}^B(B) = 1$.