Geometric Intuition of Definition of Intersection Multiplicity (of two algebraic curves) Hartshorne defines the intersection multiplicity of a point in the intersection of two plane algebraic curves $f(x, y) = 0$, $g(x, y) = 0$ in $\mathbb{A}^{2}$ to be the length of the $\mathcal{O}_{P}$-module $\mathcal{O}_{P}/(f, g)$.
I have seen in other definitions that $\dim_{k}\mathcal{O}_{P}/(f, g)$ is used instead of the length, which I guess is the same in most cases. 
Question: Does the above definition coincide with our geometric intuition of intersection multiplicity? I have seen very few related questions to this, all of which have unsatisfactory answers in regards to my question. 
I have done some examples, such as $f(x, y) = y - x^{2}, g(x, y) = y - 2x - 1$ (the tangent line at $(1, 1)$). Then our desired ring is $\mathcal{O}_{(1, 1)}/(f, g) \cong k[x,y]_{(x-1, y-1)}/(y - x^{2}, y - 2x + 1) = k[x]_{(x-1)}/(x-1)^{2}$, which has $1, x$ as basis over $k$ and so has dimension $2$. So it works! But I have still no geometric insight from doing such an example. 
 A: $\text{}$1. The way I remember is that the intersection theory of plane curves is uniquely determined by being additive, stable under linear equivalence, and for two smooth curves intersecting transversally, it is the number of intersection points. See Chapter V, 1.1 and 1.4 of Hartshorne for the connection with the local multiplicity.
$\text{}$2. Fulton works for plane curves, whereas basically the same calculations work for nonsingular surfaces more generally. Finite length or dimension of a local ring over a field are equal, e.g. use the filtration by powers of the maximal ideal, with$$m^i/m^{i+1}$$a finite sequence of finite-dimensional vector spaces.
We want$$C_1 . C_2$$for curves of a surface to be a bilinear pairing under linear equivalence. The local definition is the thing that works. When you say intuition, you mean something you can understand without doing technical definitions. The definition as length of the local quotient$$\mathcal{O}_P/(f_1, f_2)$$depends only on $C_1$, $C_2$, whereas the intersection number equals$$(f_1 + \text{bit}).(f_2 + \text{bit})$$for $C_1$, $C_2$ moved by some tiny bit; you can arrange for the new $C_1'$ and $C_2'$ to meet transversally in a number of points equal to$$\text{dim}\,\mathcal{O}_P/(f_1, f_2),$$but then you have to prove that it is independent of the choice of $C_1'$ and $C_2'$—this is a dynamic definition.
You can get quite a long way just assuming that this dynamic definition works, but it is not so good for theoretical purposes, and to relate it to coherent cohomology.
As well as Fulton, there is quite a detailed discussion in Chapters 3 and 4 of Shafarevich.
