In the book of commutative algebra of G.Kemper (can be found here), the author has proved the following theorem :
Let $A$ be an affine algebra and let $P_{0}\subsetneq P_{1}\subsetneq \cdots\subsetneq P_{n}$ be a maximal chain of prime ideals $P_i \in \text{Spec}A$. Then $n=\dim(A/P_{0})$.
I have got some trouble in understanding the proof there, so I decided to post it again in here, please help me to solve them. Here is the proof:
We use induction on $n$. Subtituting $A$ by $A/P_{0}$ we may assume that $A$ is an affine domain and $P_{0}=0$. If $n=0$ then $P_0$ is a maximal ideal so $A$ is a field and we are done. So we may assume that $n>0$, applying lemma 1.22 yields a maximal chain $P_1/P_1\subsetneq P_2/P_1\subsetneq \cdots\subsetneq P_n/P_1$ of prime ideals in $A/P_1$. Using induction, we obtain $\dim(A/P_1)=n-1$. Using Noether normalization, we obtain $C\subseteq A$ with $A$ is integral over $C$ and $C$ is isomorphic to a polynomial ring. By the maximality of the first chain: $P_{0}\subsetneq P_{1}\subsetneq \cdots\subsetneq P_{n}$ we have $\text{ht}P_1=1$. By proposition 8.8 $C$ is normal so all hypotheses of theorem 8.17 are satisfied. We obtain $\text{ht}(C\cap P_1)=1$. By the implication $(b)\Rightarrow (a)$ of theorem 5.13, this implies $\dim(C/C\cap P_1)=\dim C-1$. Since $A/P_1$ is integral over $C/C\cap P_1$ then corrollary 8.13 yields : $\dim(A/P_1)=\dim(C/C\cap P_1)=\dim C-1=\dim A-1$.
Here are my questions:
- How can we subtitute $A$ by $A/P_0$ ?
- What kind of induction gives us the result $n-1=\dim A/P_1$ ?
Edit : In response to YACP's comment, here is my idea of induction that I feel that it does not fit in this case : Suppose the hypothesis of theorem is true for all $n\le k$, i.e we have $P_{0}\subsetneq P_{1}\subsetneq \cdots\subsetneq P_{k}$ is a maximal chain of primes ideals in $A$, then $k=\dim(A/P_0)$. Now we will prove that proposition for $n=k+1$, i.e if $P_{0}\subsetneq P_{1}\subsetneq \cdots\subsetneq P_{k}$ is a maximal chain of prime ideals in $A$, then $\dim(A/P_0)=k+1$. What am I wrong? I still do not understand the idea of induction giving us the result $n-1=\dim A/P_1$.
