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Hi, due to Nagata and his clever and bizzare examples I'm unsure in this: 1) Is there a regular ring of infinite Krull dimension? 2) Is it true that: Regular ring of finite Krull dimension = Commutative Noetherian ring of finite global dimension? (answer is yes if R is local by Serre's Theorem) Regards, David |
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For 1), Nagata's counterexample goes as follows: Let $\mathbb{N}=A_1\cup A_2 \cup \cdots$ be a partition such that $|A_i|<|A_{i+1}|$. Take $S=k[x_1,x_2,\ldots]$, prime ideals $I_i=\langle x_i | i\in A_i \rangle$ and consider the localization $S[U^{-1}]$ in the multiplcative set $U=S \setminus \bigcup_i I_i$. Then the height of each of the $I_i$ is $|A_i|$ and so $S[U^{-1}]$ has infinite Krull dimension. It not so hard to show that the ring is regular, but a more surprising point is that it is also Noetherian. For 2) I believe the answer is yes by this reference. |
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