2

Let $A$ be a Noetherian domain, and consider $A[X,Y]$. Let $p\subset A[X,Y]$ be a prime ideal, so that $B:=A[X,Y]/p$ is (a domain and) an $A$-algebra of finite type. Now, it is known this result:

If $R$ is a $k$-algebra of finite type that is a domain, $\operatorname{dim}R=\operatorname{trdeg}_k \operatorname {Frac}R $.

Is there a way to find the Krull dimension of $B$ (using $\operatorname{dim}A$ as a known constant) that relies on such result?

The main difficult is that I don't find any algebra over any field in this situation; the only one is $\operatorname{Frac}B$ that is a $\operatorname{Frac}A$-algebra, but it isn't finitely generated a priori, so we can't use the result above. Is that theorem just unrelated to this situation or I'm not seein how to use it?

Dr. Scotti
  • 2,614
  • -1: This is really under-specified - without more assumptions, there is no hope of any reasonable result. For instance, $\dim A$ could be anything (even infinite), $\dim A[X,Y]$ need not be $\dim A+2$, and $\dim B \leq \dim A[X,Y]$ with any possibility attained. What are you really trying to do here? – Hank Scorpio Jan 22 '22 at 23:57
  • @HankScorpio sorry I actually missed an hypothesis, that is $A$ Noetherian. If you want to know from what concrete problem came this question, you can look at this question (https://math.stackexchange.com/questions/4353366/compute-krull-dimension); in particular the first comment seems an hint related to this theorem – Dr. Scotti Jan 23 '22 at 06:36
  • In the linked thread you want to find the dimension of $B$ (in the actual notation) knowing the dimension of $A$. Here you know nothing and want to find the dimension of $A$. (-1) – user26857 Jan 23 '22 at 07:41
  • @user26857 of course you are right, I corrected the post – Dr. Scotti Jan 23 '22 at 07:45
  • This is not as much as a fix as you might believe: there are infinite-dimensional noetherian domains, and still all you know is that $\dim B \leq \dim A[X,Y]$ with all values possible! Take $A=k[t_1,\cdots,t_n]$ and $p=(t_1,\cdots,t_m)$ to get an example with Krull dimension $n+2-m$, for instance. The result you list is also not useful in general, since $Frac B$ need not admit a map from $Frac A$ - consider $B$ a nontrivial quotient of $A$, for instance. I'm voting to close this to give you a chance to write a question that has a good answer. – Hank Scorpio Jan 23 '22 at 08:36

0 Answers0