Intersection multiplicity

Remarks

The sum is finite because $A$ is a regular local ring, and each term has finite length because $Z$ is a component of $V \cap W$. This is Serre's Tor formula, and $i(Z; V \cdot W) \ge 1$. When $A/I_V$ and $A/I_W$ are Cohen–Macaulay, the higher Tor terms vanish and $i(Z; V \cdot W) = \ell_A\big(A/(I_V + I_W)\big)$. For two plane curves $f = 0$ and $g = 0$ without common component, meeting at a point $P$, this gives $i_P(f, g) = \dim_k \mathcal{O}_{\mathbb{A}^2, P}/(f, g)$. Example: for $f = y - x^2$ and $g = y$ at $P = (0, 0)$, $\mathcal{O}_{\mathbb{A}^2, P}/(y - x^2, y) \cong k[x]_{(x)}/(x^2)$, of dimension $2$. Non-example: if $V = W$ is a line in $X = \mathbb{A}^2$, then $V \cap W$ has dimension $1$, not $1 + 1 - 2 = 0$, so the intersection is not proper and the multiplicity is not defined.