Problems with a theorem on Push-forward of cycles I am studying the book Intersection Theory by Fulton, reading it for the first time and having some problems with the first proposition that appears on this book. 
There are some definitions that have been given: 
1) Let $X$ a variety, $V$ a subvariety of $X$ of codimension one. The local ring $A=\mathcal{O}_{V,X}$ is one dimensional local domain. Let $r\in R(X)^*$, we define an homomorphism $ord_V:R(X)^*\rightarrow \mathbb{Z}$ with $ord_V(r):= l_A(A/(r))$, where $l_A$ denotes the length of the $A$-module in parentheses.
2) For any $(k+1)$-dimensional subvariety $W$ of $X$, and any $r\in R(W)^*$, define a k-cycle $[div(r)]$ on $X$ by $[div(r)]=\sum ord_V(r) [V]$ the sum over all codimension one subvariety $V$ of $W$.
3) Let $f:X\rightarrow y$ be a proper morphism. For any subvariety $V$ of $X$, the image $W=f(V)$ is a closed subvariety of $Y$. Define $f_*[V]= deg(V/W)[W]$, where $deg(V/W)=[R(V):R(W)]$ is the degree of the field extension if $dim(W)=dim(V)$ and $deg(V/W)=0$ if $dim(W)<dim(V)$.
$f_*$ extends linearly to a homomorphism $f_*:Z_kX\rightarrow Z_kY$, 
Proposition: Let $f:X\rightarrow Y$ be a proper, surjective morphism of varieties, and let $r\in R(X)^*$. Then
a) $f_*[div(r)]=0$ if $dim(Y)<dim(X)$.
b) $f_*[div(r)]=[div(N(r))]$ if $dim(Y)=dim(X)$.
Where $R(X)$ is a finite extension of $R(Y)$, and $N(R)$ is the norm of $r$, i.e., the determinant of the $R(Y)$-linear endomorphism of $R(X)$ given by multiplication by $r$.
There is given a proof of the proposition, separated by cases.
On case $1$: $Y=Spec(k)$, $k$ a field, $X=\mathbb{P}_{k}^{1}$, we can assume $r$ is an irreducible polynomial of degree $d$, in $k[t]$.Then $r$ generates a prime ideal $p$ in $k[t]$ corresponding to a point $P$ in $X$ with $ord_P(r)=1$. Then he makes a lot of affirmations that I cannot follow:
1) The only other point along which $r$ has non-zero orderis the point $P_{\infty}=(0:1)$ at infinity. I don't understand this completely, I computed $ord_Q(r)=l_{\mathcal{O}_{Q,X}}(\mathcal{O}_{Q,X}/(r))=l_{\mathcal{O}_{Q,X}}(0)=0$ for all $Q$ different of $P$, is that correct? And I cannot compute $ord_{\mathbb{P}_{\infty}}(r)$, because I cannot figure out what is $l_{\mathcal{O}_{\mathbb{P}_{\infty},X}}(\mathcal{O}_{\mathbb{P}_{\infty},X}/(r))$.
2) Making $s=1/t$ a uniformizing parameter, then $s^dr$ is a unit at $\mathbb{P}_{\infty}$ (what is a unit at $\mathbb{P}_{\infty}$?), then $ord_{\mathbb{P}_{\infty}}(r)=-d$ (again, why? I know that $ord$ is a homomorphism, but even with that I canot finish).
Appologize for the big text, but all the definitions are new for me, so I hope anyone can give me some ideas on this, thanks.
 A: When Fulton said that he's assuming that $r$ is a polynomial, this means really that the restriction to $\Bbb A^1 \subset \Bbb P^1$ is a polynomial (because they are no polynomial functions defined on $\Bbb P^1$, only rational functions). So this means that $r = f(x)/y^d$ where $x,y$ are the usual coordinates on $\Bbb P^1$. I m not sure to understand exactly what Fulton do mean here, but let's try to understand the situation : we can write $$ r(x,y) = \frac{(x-a_1y)^{m_1} \dots  (x-a_ky)^{m_k}}{y^d}$$
For example, if we look at $P_1 = [a_1:1]$, in the local ring $\mathcal O_{\Bbb P^1,P_1}$ $x-a_iy$ are units for $i = 2, \dots, k$ and $y^{-d}$ also so really $\text{ord}_{P_1}(r) = m_1$. This gives $\text{ord}_{P_i}(r) = m_i$ and $\text{ord}_{Q}(r) = -d$ where $Q = [1:0]$. 
I hope this makes everything a bit clearer. Also, the book of Fulton has the reputation to be difficult to read, a more gentle book about  intersection theory is 3264 and all that, by Eisenbud and Harris. They don't provide a proof of every theorem but provide very good motivation and a great class of examples, definitely worth looking. 
