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 $\Omega_{\mathcal{X}}^+$ of the sheaf of differentials $\Omega_{\mathcal{X}}$. It is defined in a similar way as the subsheaf $\mathcal{O}^+_{\mathcal{X}}$ of $\mathcal{O}_{\mathcal{X}}$ using K\"ahler seminorms on $\Omega_{\mathcal{X}}$. We give a description of $\Omega^+_{\mathcal{X}}$ 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 $\Omega^+_{\mathcal{X}}(\mathcal{X})$ is isomorphic to the logarithmic differentials of $(X,\bar{X})$.