Logarithmic differentials on discretely ringed adic spaces

Abstract

On a smooth discretely ringed adic space $\mathcal{X}$ over a field $k$ we define a subsheaf $\hat{I}{\text{c◯}}_{\mathcal{X}}^{+}$ of the sheaf of differentials $\hat{I}{\text{c◯}}_{\mathcal{X}}$. It is defined in a similar way as the subsheaf ${\mathcal{O}}_{\mathcal{X}}^{+}$ of ${\mathcal{O}}_{\mathcal{X}}$ using Kähler seminorms on $\hat{I}{\text{c◯}}_{\mathcal{X}}$. We give a description of $\hat{I}{\text{c◯}}_{\mathcal{X}}^{+}$ in terms of logarithmic differentials. If $\mathcal{X}$ is of the form $\mathrm{Spa}(X,\bar{X})$ for a scheme $\bar{X}$ and an open subscheme $X$ such that the corresponding log structure on $\bar{X}$ is smooth, we show that $\hat{I}{\text{c◯}}_{\mathcal{X}}^{+}(\mathcal{X})$ is isomorphic to the logarithmic differentials of $(X,\bar{X})$.