The Stacks project

Remark 31.16.3. It is not clear how to characterize the non-Noetherian local rings $(A, \mathfrak m)$ whose punctured spectrum is affine. Such a ring has a finitely generated ideal $I$ with $\mathfrak m = \sqrt{I}$. Of course if we can take $I$ generated by $1$ element, then $A$ has an affine puncture spectrum; this gives lots of non-Noetherian examples. Conversely, it follows from the argument in the proof of Lemma 31.16.1 that such a ring cannot possess a nonzerodivisor $f \in \mathfrak m$ with $H^0_ I(A/fA) = 0$ (so $A$ cannot have a regular sequence of length $2$). Moreover, the same holds for any ring $A'$ which is the target of a local homomorphism of local rings $A \to A'$ such that $\mathfrak m_{A'} = \sqrt{\mathfrak mA'}$.