What are some examples of Tate rings $R$ (i.e. Huber rings with with topologically nilpotent units) which are Noetherian but *not* strongly Noetherian ($R$ is strongly Noetherian iff for all $n \in \mathbb{N}$, the corresponding $R$-algebra of convergent power series $R\{x_1, ..., x_n\}$ is Noetherian) ?

## 1 Answer

The only example I know of occurs in *On Hausdorff completions of commutative rings in rigid geometry* by Fujiwara, Gabber, Kato (and according to the intro the example is due to Gabber). The example is the subject of $\S$8.3, in their language they give an example of a $t$-adically complete ring $A_0$ which is Noetherian outside $(t)$ but $A_0\langle x\rangle$ is not noetherian outside $(t)$. In this setting $A:=A_0[1/t]$ has a unique structure of a Tate-Huber ring making $A_0$ a ring of definition, and then $A$ being Noetherian is the same as $A_0$ being Noetherian outside $(t)$, so one gets a Noetherian Tate ring $A$ for which $A\langle x\rangle$ is not Noetherian. In addition they prove in Lemma 8.3.3 that $A$ is in fact a field (but of course not a nonarchimedean field).