Logarithmic Differentials on Adic Spaces

Abstract: The object of interest in this talk is a certain subsheaf $\Omega^+_X$ of the sheaf of differentials $\Omega_X$ of a discretely ringed adic space $X$ over a field $k$. The first part will be dedicated to an introduction to discretely ringed adic spaces. We will then define $\Omega^+_X$ using K\"ahler seminorms and establish a relation with logarithmic differentials. Finally we study the case where $X = Spa(U,Y)$ for a scheme $Y$ over $k$ and a subscheme $U$ such that the corresponding log structure on $Y$ is log smooth. It turns out that $\Omega^+_X(X)$ equals $\Omega^{log}_{(U,Y)}(U,Y)$.

Mathematics

Audience: researchers in the topic

( paper )

Emmy Noether Kolloquium Mainz