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> Does there exist a local domain of infinite dimension in which every chain of prime ideals is finite?
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Of course, such a ring must be neither noetherian nor catenary.


(This question arose while trying to compare different definitions of catenarity, and more precisely while trying to understand what may cause codimensions in topological spaces to be infinite.)



An infinite dimensional local domain whose chains of primes are finite
Choose a field $k$ and a Noetherian $k$-algebra $R$ of infinite dimension. (I know you know such a thing exists.) Let $R' \subset R[x]$ be the set of polynomials $f = \sum a\_i x^i$ whose constant term is constant, i.e., $a\_0 \in k \subset R$. Then we have
$$
\text{Spec}(R') = \text{Spec}(R[x]) \amalg\_{\text{Spec}(R)} \text{Spec}(k)
$$
by [Tag 0B7J](https://stacks.math.columbia.edu/tag/0B7J). Observe that $R'[1/x] = R[x, 1/x]$ is Noetherian. Observe that for any prime $\mathfrak p \subset R$ the prime $\mathfrak p' = R' \cap \mathfrak p R[x]$ is contained in the maximal ideal $\mathfrak m = \text{Ker}(R' \to k) = \sqrt{xR'}$ (small detail omitted). Thus the local ring of $R'$ and $\mathfrak m$ is an example. Namely, its punctured spectrum is Noetherian of infinite dimension (as the localization of $R'$ at $\mathfrak p'$ is the same as the localization of $R[x]$ at $\mathfrak p R[x]$ which is flat over $R\_\mathfrak p$ and hence has dimension $\geq \dim(R\_\mathfrak p)$. Enjoy!


