My question is a generalization of Analytic and algebraic definitions of intersection multiplicity of two complex algebraic curves coincide.
Take two complex space germs $(A, 0)=V(I_A)$ of dimension $a$ and $(B, 0)=V(I_B)$ of dimension $b$ in $\mathbb{C}^{a+b}$, where $I_A$ and $I_B$ are ideals of $\mathcal{O}_{\mathbb{C}^{a+b}, 0}$. Consider the algebra $$ \frac{\mathcal{O}_{\mathbb{C}^{a+b}, 0}}{I_A +I_B},$$ whose dimension is a candidate of the intersection multiplicity of $A$ and $B$ at $0$.
First question: In which cases and in which sense is it really the intersection multiplicity? (E.g. complete intersections, Cohen--Macaulay property...)
Assume that we have a parametrization $\phi: (\mathbb{C}^a, 0) \to (\mathbb{C}^{a+b}, 0)$ of $(A, 0)=\mbox{im}(\phi)$. Analogously with the case of curves, another candidate for the intersection multiplicity is the dimension of the algebra $$ \frac{\mathcal{O}_{\mathbb{C}^{a}, 0}}{\phi^*(I_B) \cdot \mathcal{O}_{\mathbb{C}^{a}, 0}}.$$
Second question: Is it true that the two candidates are equal to each other? Are the two algebras isomorphic?
I tried to prove the isomorphism starting from the observation $$I_A=\ker (\phi^*: \mathcal{O}_{\mathbb{C}^{a+b}, 0} \to \mathcal{O}_{\mathbb{C}^{a}, 0}),$$ yet unsuccessfully. I am interested in references as well.
