# Differential forms in positive characteristic II: cdh-descent via functorial Riemann-Zariski spaces

## TL;DR

This paper advances the understanding of sheaves associated with K"ahler differentials in positive characteristic, establishing cdh-descent without resolution of singularities and linking to seminormalisation and potential connections to Berkovich spaces.

## Contribution

It provides a calculation of cdh-sheaves via seminormalisation and shows the equivalence between cdh-sheaves and seminormal varieties, extending the theory without resolution assumptions.

## Key findings

- Calculation of cdh-sheaves using seminormalisation
- Equivalence between cdh-sheaves and seminormal varieties
- Potential connections to Berkovich spaces and F-singularities

## Abstract

This paper continues our study of the sheaf associated to K\"ahler differentials in the cdh-topology and its cousins, in positive characteristic, without assuming resolution of singularities. The picture for the sheaves themselves is now fairly complete. We give a calculation $\mathcal{O}_{cdh}(X) \cong \mathcal{O}(X^{sn})$ in terms of the seminormalisation. We observe that the category of representable cdh-sheaves is equivalent to the category of seminormal varieties. We conclude by proposing some possible connections to Berkovich spaces, and $F$-singularities in the last section. The tools developed for the case of differential forms also apply in other contexts and should be of independent interest.