Une caractérisation de ${\mathbb{B}}_{\mathrm{d}\mathrm{R}}^{+}$ et $\mathcal{O}{\mathbb{B}}_{\mathrm{d}\mathrm{R}}^{+}$

We show that ${\mathbf{B}}_{\mathrm{d}\mathrm{R}}^{+}$ is the universal thickening of ${\mathbf{C}}_p$. More generally, we show that, if $S$ is a reduced affinoid algebra, $\mathcal{O}{\mathbb{B}}_{\mathrm{d}\mathrm{R}}^{+}(\bar{S})$ is the universal $S$-thickening of the completion of $\bar{S}$.