The Stacks project

Example 85.9.2. Let $(A, \mathfrak m, \kappa )$ be a valuation ring, which is $(\pi )$-adically complete for some nonzero $\pi \in \mathfrak m$. Assume also that $\mathfrak m$ is not finitely generated. An example is $A = \mathcal{O}_{\mathbf{C}_ p}$ and $\pi = p$ where $\mathcal{O}_{\mathbf{C}_ p}$ is the ring of integers of the field of $p$-adic complex numbers $\mathbf{C}_ p$ (this is the completion of the algebraic closure of $\mathbf{Q}_ p$). Another example is

\[ A = \left\{ \sum \nolimits _{\alpha \in \mathbf{Q},\ \alpha \geq 0} a_\alpha t^\alpha \middle | \begin{matrix} a_\alpha \in \kappa \text{ and for all }n\text{ there are only a} \\ \text{finite number of nonzero }a_\alpha \text{ with }\alpha \leq n \end{matrix} \right\} \]

and $\pi = t$. Then $X = \text{Spf}(A)$ is an affine formal algebraic space and $\mathop{\mathrm{Spec}}(\kappa ) \to X$ is a thickening which corresponds to the weak ideal of definition $\mathfrak m \subset A$ which is however not an ideal of definition.

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