I'm interested in relating the multiplicity of the intersection of curves defined over a local to the corresponding curves in the residue field.
Let $A$ be a DVR with maximal ideal $m$ and $f_1,f_2 \in A[x,y]$, which reduce to polynomials $\bar f_1,\bar f_2$ in $(A/m)[x,y]$. If $P = (p_1,p_2)$ is a point on the intersection of the curves corresponding to $f_1$ and $f_2$, let $mul(P)$ denote the multiplicity of that intersection (the length of $K[x,y]/(f_1,f_2)$ localized at $(x-p_1,y-p_2)$).
Is it the case that for a point $Q$ in the intersection of the curves defined by $\bar f_1$ and $\bar f_2$ over $A/m$ that the multiplicity of $Q$ is $$\sum_{P \textrm{ reducing to } Q} mul(P)$$
Even if it's not generally true but holds under some extra hypotheses, I would be very interested. It seems like it should reduce to simple commutative algebra but I've been stuck for a while. A reference would be great too!