Intersection multiplicity , Hilbert samuel polynomials and normal cones I'm in the process of reading some parts of Fulton's "Introduction to intersection theory" and there's a short part there which I don't quite understand where I think I'm missing something obvious.  
Background 
Let us assume that we work over $\mathbb{C}$ for simplicity, and let $H_1, \ldots, H_n$ be hypersurfaces in some n-dimensional variety $V$ and let $p$ be an isolated point of $\cap H_i.$ Let $A= \mathcal{O}_{V,p}$ be the local ring of $V$ along $P$ and assume that each $H_i$ is defined by tone element $f_i$ in $A.$  Let $I=(f_1,\ldots,f_n).$ Then $A/I$ is of (Krull) dimension zero. One defines the multiplicity $i(P,H_1\cdots H_n)$ to be the coefficient of $t^n/n!$ in the Hilbert-Samuel polynomial $P(t) = \text{dim }_{\mathbb{C}} (A/I^t) $ (so $P(t)$ is the eventual polynomial for that function). 
The projective normal cone and the question
Set $\Lambda = A/I$ and consider the surjection of graded rings $$\Lambda[X_1,\ldots,X_n] \rightarrow \oplus_{t=0}^{\infty} I^t/I^{t+1}$$ 
taking $X_i$to $f_i \in I/I^2$. The kernel of this homomorphism is a homogenous ideal which defines a subscheme $\mathbf{P}(C)$ of $P^{n-1}_{\Lambda}.$ This is what is known as the projective normal cone. Then, one can see that the degree of $\mathbf{P}(C)$ in $\mathbb{P}^{n-1}_{\Lambda}$, that is, the coefficient of $t^{n-1}/(n-1)!$ in the Hilbert polynomial $P_{\mathbf{P}(C)}(t)$, is precisely the multiplicity defined earlier. This is not hard to show. However:  
The claim is that the degree of $P_{\mathbf{P}(C)}(t)$ is the same as the length of $\mathbf{P}(C)$ at the generic point. I can't for the life of me seem to figure out why this should be true, so I would be very grateful for some help.
 A: $\newcommand{\ideal}[1]{{\mathfrak #1}}$
$\newcommand{\PSP}{{\mathbf P}}$
$\newcommand{\proj}[1]{{\mathrm{proj}}(#1)}$
$\newcommand{\Ass}{\mathrm{Ass}}$
$\newcommand{\supp}{\mathrm{supp}}$
$\newcommand{\length}{\mathrm{length}}$
Consider the sequence
$$
0 \to J \to R = \Lambda[x_1,\ldots,x_n] \to \bigoplus_t I^t/I^{t+1} = U \to 0
$$
so $S = R/J$ is isomorphic to $U$. Note that $\PSP(C) = \proj{R/J}$ and $\dim R/J = n$
Let $\bar{\ideal{m}}$ be the maximal ideal of $A/I$. Then
$\ideal{p} = \bar{\ideal{m}}[x_1,\ldots,x_n]=\bar{\ideal{m}}R$ is a prime ideal of $R$ and consists only of nilpotent elements. So it is also the nilradical $\ideal{N}$ of $R$.
First we show, that $J \subseteq \ideal{p} \subseteq R$. It is
$$n = \dim R/J = \dim R/(J+\ideal{N})$$
as every prime $\ideal{q}$ of $R$ that contains $J$ also contains $\ideal{N}$.
But if $J + \ideal{N} \supsetneq \ideal{N}$, then as $R/\ideal{N}=R/\ideal{p}$ is the integral affine algebra $A/\ideal{m}[x_1,\ldots,x_n]$, we would necessary have $\dim R/(J+\ideal{N}) < \dim R/\ideal{N} = n$.
So $V(J)$ is an affine algebraic scheme with underlying variety $V(\ideal{p})$.
So it makes sense to consider the length of $(R/J)_{\ideal{p}}$ that you call "the length of $\PSP(C)$ at the generic point" and that we abbreviate now $i(P)'$.
Now set $M_m$ be the $R$-module $R/J$ and consider the typical filtration from the theory of associated primes
$$(*) \quad 0 \to M_{i-1} \to M_i \to R/\ideal{p}_i \to 0$$
where $\ideal{p}_i \in \Ass R/J$ with $\ideal{p} = \min \Ass R/J = \supp R/J$ occuring - and that will be shown next - exactly $i(P)'$ times.
Now simply consider all sequences $(*)$ tensored by $\otimes_R R_\ideal{p}$
to give
$$(**) \quad 0 \to (M_{i-1})_\ideal{p} \to (M_i)_\ideal{p} \to (R/\ideal{p}_i)_\ideal{p} \to 0$$
The $(R/\ideal{p}_i)_\ideal{p}$ are of length $1$ exactly for $\ideal{p} = \ideal{p}_i$ otherwise the localized module itself vanishes.
So as $\length((M_m)_\ideal{p}) = \length (R/J)_\ideal{p} = i(P)'$ we have $\ideal{p}$ occuring $i(P)'$ times in the filtrations $(*)$.
But now consider the Hilbert-polynomials $P_{M_i}$ and $P_{R/\ideal{p}_i} = Q_i$
from the sequences $(*)$. We have the relation
$$(***) \quad P_{M_i} = P_{M_{i-1}} + Q_i$$
The only $Q_i$ that are of degree $n$ are those where $\ideal{p} = \ideal{p}_i$, which occur, as shown above, $i(P)'$ times.
Furthermore for these because of $R/\ideal{p} = A/\ideal{m}[x_1,\ldots,x_n]$:
$$Q_i = t^n/n! + \text{lower terms}$$
So alltogether we get
$$P(t) = P_{R/J} = P_{M_m} = i(P)' t^n/n! + \text{lower terms}$$
As we knew a priori $P(t) = i(P)t^n/n! + \text{lower terms}$ (see Fulton, page 11, (i)) we have the equality $i(P) = i(P')$ which relates the Hilbert-Samuel multiplicity with the one that Fulton introduced earlier on page 9 of the "Introduction".
Note that the key idea of using the $\Ass$-filtration of $R/J$ and localizing by $\ideal{p}$ is exactly used in the same way as in Hartshornes proof of the Bezout-Theorem.
