Global intersection length is the sum of the local multiplicities

Given: AC, plane curves $C=V(F)$, $D=V(G)$ over the algebraically closed field $k$ with no common component, and $X=\mathrm{Proj}(k[x_0,x_1,x_2]\mathrm{/}(F,G))$ Projective scheme of a homogeneous quotient and its standard affine charts.

$X$ is zero-dimensional with finitely many points, and $X$'s points correspond bijectively to the points of $C\cap D$: a point $x\in X$ lies in a standard chart $D_{+}(x_i)$, whose chart ring is $k[u,v]\mathrm{/}(f_i,g_i)$ for the dehomogenised forms, and the maximal ideals of that chart ring are the evaluation ideals at the common zeros of $f_i,g_i$ A plane intersection with no common component is nonempty and zero-dimensional, A zero-dimensional projective scheme has finitely many closed points with finite-dimensional local rings, Prime and local-ring correspondence on standard projective charts, Two coprime projective plane forms meet in total length equal to their degree product, The residue field at a point of an affine scheme. AC enters through these published suppliers.

The local ring of $X$ at a point $x_p$ corresponding to $p$ is ${\mathcal{O}}_{X,x_p}\cong {\mathcal{O}}_{{\mathbf{P}}^2,p}\mathrm{/}(f_p,g_p)$: in a chart containing $p$ the chart ring is $k[u,v]\mathrm{/}(f_i,g_i)$ and ${\mathcal{O}}_{X,x_p}$ is its localisation at the prime of $p$; localisation commutes with the quotient, and the localisation of $k[u,v]$ at $p$ modulo the dehomogenised equations is ${\mathcal{O}}_{{\mathbf{P}}^2,p}$ modulo the local ideal Two coprime projective plane forms meet in total length equal to their degree product, Localisation commutes with quotient rings: $S^{-1}R\mathrm{/}{S}^{-1}I\cong {\bar{S}}^{-1}(R\mathrm{/}I)$, A localisation is unique up to a unique isomorphism compatible with the map from $R$, Universal property of localisation: maps that invert $S$ factor uniquely through $S^{-1}R$.

The total length is the finite weighted sum ${\mathrm{len}}_k(X)={\sum }_{x\in X}{\mathrm{\ell }}_{{\mathcal{O}}_{X,x}}({\mathcal{O}}_{X,x})[\kappa (x):k]$ Total length of a zero-dimensional projective scheme, and over the algebraically closed field every residue degree is one, $\kappa (x)=k$ A field is algebraically closed exactly when every nonconstant polynomial splits, equivalently when it has no nontrivial finite extension, An algebraically closed field: every nonconstant polynomial has a root in the field.

Since $C,D$ have no common component, they share no local branch at any point, so each $I_p(C,D)$ is finite by the finiteness lemma Finite local length exactly when no common local branch; the sum over the finite set $C\cap D$ is therefore a finite sum of finite numbers.