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This theorem is from Matsumura (p.34)

Let $k$ be a field and $A$ an integral domain which is finitely generated over $k$. Then $\dim A = \operatorname{trdeg}_k A$ (where $\operatorname{trdeg}_k A$ is the transcendence degree of the field of fractions of $A$ over $k$).

I've been trying to read this proof but there are so many things that are confusing me, so I would appreciate any kind of help.

Proof

Let $A = k[X_1, ... , X_n]/P$, and set $r = \operatorname{trdeg}_k A$

Question 1: From what I understand $A = k[a_1, ... , a_n]$ for some $a_1, ... , a_n$ since it is finitely generated over $k$, and we have $f: k[X_1, ... ,X_n] \rightarrow k[a_1, ... , a_n]$ sending $X_i$ to $a_i$. So the first isomorphism theorem gives us $k[X_1, ... , X_n]/\ker f \simeq k[\alpha_1, ... ,\alpha_n]$. Since $A$ is an integral domain, $\ker f$ must be prime and we just call it $P$, right?

Question 2: Since $A$ is a module over $k$, the dimension of $A$ is defined to be the Krull dimension of $k/\operatorname{ann}(A)$, right?

To prove that $r \geq \dim A$ it is enough to show that if $P$ and $Q$ are prime ideals of $k[X] = k[X_1, ... ,X_n]$ with $Q \supset P$ and $Q \not= P$, then $$\operatorname{trdeg}_k k[X]/Q < \operatorname{trdeg}_k k[X]/P.$$

Question 3: Why would it be enough to show that? How does that say anything about the Krull dimension of $k/\operatorname{ann}(A)$?

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  1. True.

  2. False: $\operatorname{ann}_k(A)=0$. The considered Krull dimension is that of $A$ as a commutative ring.

  3. Let's say $n=\dim A$. Then there is a chain of prime ideals $(0)=p_0\subset\cdots\subset p_n$ in $A$. Now note that $p_i=P_i/\ker f$, where $P_i$ is a prime ideal in $k[X]$. Moreover, $A/p_i=k[X]/P_i$. In particular, $P_0=\ker f$ and $A=k[X]/P_0$. From the inequality mentioned by the proof we get $$\operatorname{trdeg}_k k[X]/P_n+n\le \operatorname{trdeg}_k k[X]/P_0=\operatorname{trdeg}_kA.$$ Now use that $\operatorname{trdeg}_k k[X]/P_n=0$ (why?).

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  • $\begingroup$ @user26857 $tr \deg_k k[X]/P_n = 0$ because $P_n$ is maximal, meaning that $P_n = (X)$...so we have $tr \deg_k k = 0$. But I have one more question (if you don't mind). The inequality tells us that $tr \deg k[X]/P_n \leq tr \deg k[X]/P_0$, right? It does not tell us that $tr \deg k[X]/P_n + n \leq tr \deg_k k[X]/P_0$. $\endgroup$ – Artus Oct 18 '14 at 18:41
  • $\begingroup$ @Artos In your question it is written $<$, not $\le$, right? Or if $a,b$ are integers and $a<b$ then $a+1\le b$. (Btw, $P_n$ is maximal, but isn't necessarily $(X)$.) $\endgroup$ – user26857 Oct 18 '14 at 18:44
  • $\begingroup$ @user26857 Oh...I'm sorry, I forgot about the chain length...thanks $\endgroup$ – Artus Oct 18 '14 at 18:48

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