Let $I$ be an ideal of $R$. An element $r\in R$ is integral over $I$ if $r$ satisfies a relation of the form $$r^n+a_1r^{n-1}+\ldots+a_{n-1}r+a_n=0,$$ where $a_i\in I^i$ for every $i=1,\ldots,n$. The set of integral elements over $I$ is called the integral closure of $I$ and denoted by $\overline{I}$.
I'd like to prove that the integral closure of a monomial ideal is still a monomial ideal: the subsequent proof is taken from "Integral closure of ideals, rings and modules" of I. Swanson and C. Huneke:
I don't understand the last half of the proof (from the point of the definition of $g$): I see that a wise choice of $u_1,\ldots,u_d$ lead to $\phi_u(f)$ is not a multiple of $f$, but from that point on I'm kind lost.
- what are $L_1,\ldots,L_d$, why do we need to power some costants $u_i$ to $u_i^{L_{d_i}}$
- why is the degree $L$ component of $g$ is $0$?
- in the last lines I'm kinda lost, I simply don't see why its statement are true
I'd really like to understand the strategy of this proof: I did not find any other reference for this because everyone says it is a well known result without actually giving at least a reference and I'd like to understand it. Any help would be much appreciate, thanks in advance!
The proof implicitly uses the notation $L=(L_1,\ldots,L_d)$ for the multi-index $L$ where $f_L\ne0$. Moreover, we simply have $f_l=c_l\cdot X^{l_1}\cdots X^{l_d}$ for any $l\in\Lambda$, where $c_l\in k$ are certain constants With this notation, \begin{align} g &= u_1^{L_1}\cdots u_{d}^{L_d}\cdot f - \varphi_u(f) \\&=\left(\sum_{l\in\Lambda} u_1^{L_1}\cdots u_{d}^{L_d}\cdot c_l\cdot X_1^{l_1}\cdots X_d^{l_d}\right) - \left(\sum_{l\in\Lambda} u_1^{l_1}\cdots u_{d}^{l_d}\cdot c_l\cdot X_1^{l_1}\cdots X_d^{l_d}\right) \\&=\sum_{l\in\Lambda} \left(u_1^{L_1}\cdots u_{d}^{L_d} - u_1^{l_1}\cdots u_{d}^{l_d}\right) \cdot c_l\cdot X_1^{l_1}\cdots X_d^{l_d} \end{align} Now you can see that $g$, unlike $f$, is zero in degree $L$ because the term $$ \left(u_1^{L_1}\cdots u_{d}^{L_d} - u_1^{l_1}\cdots u_{d}^{l_d}\right) $$ becomes zero for $l=L$.