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We know that the affine spectrum of a local ring is a connected doubleton, i.e. topologically it is the set \(\{p,q\}\) with open sets \(\{\emptyset, \{p\}, \{p,q\}\}\). Now consider the topological space with underlying set \(\{p,q,r\}\) with open sets \(\{\emptyset, \{p\}, \{p,q\}, \{p,q,r\}\}\). Is there a way to realise this as an affine scheme?

If we think about \(\mathbb{C}[X,Y]_{(X,Y)}\), this has the right number of primes at heights \(0\) and \(2\) but way too many at height \(1\), so we need to remove those. However, we can’t do this with the usual operations of localization and quotients– something more exotic is needed. The trick is to consider the subring \[R:=\{f(X,Y)\in \mathbb{C}[X,Y^\pm]\;|\; f(0,Y)\in \mathbb{C}[X,Y]\}\] of \(\mathbb{C}[X,Y^\pm].\) In other words, we allow inverting \(Y\) but only if it appears with an \(X\).

The ideal \(\mathfrak{m}=(X,Y)\subset R\) is maximal, and \(0\) is prime. Suppose we have \(0\subsetneq \mathfrak{p}\subsetneq \mathfrak{m}\) for some prime ideal \(\mathfrak{p}\), then it must have terms with no positive power of \(Y\) (by multiplying with \(XY^{-n}\) for \(n\) sufficiently large). Choose \(f\in \mathfrak{p}\) such that \(f\in \mathbb{C}[X,Y^{-1}]\) and \(\text{deg}_Xf\) is minimal among these. Since \(f\in (X,Y)\cap \mathbb{C}[X,Y^{-1}]\), we must have \(f=X\cdot g\) for some \(g\in \mathbb{C}[X,Y^{-1}]\). But then \(\text{deg}_Xg<\text{deg}_Xf\) hence \(g\notin \mathfrak{p}\), so \(X\in \mathfrak{p}\) i.e. \(\mathfrak{p}=(X)\). Thus the localisation \(R_\mathfrak{m}\) has the required spectrum, with exactly one prime at heights \(0\),\(1\) and \(2\) each.

There is a way to state and extend this example using valuation rings, on totally ordered groups other than \(\mathbb{Z}\)– see Hahn series rings. The moral to me is that there are more exotic rings than I know of!