#### Vol. 12, No. 3, 2018

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

### Annette Huber and Shane Kelly

Vol. 12 (2018), No. 3, 649–692
##### Abstract

This paper continues our study of the sheaf associated to Kähler 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 ${\mathsc{O}}_{cdh}\left(X\right)\cong \mathsc{O}\left({X}^{sn}\right)$ 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.

##### Keywords
differential forms, cdh-topology, valuation rings, seminormalization, singularities
Primary: 14G17
Secondary: 14F20
##### Milestones
Received: 5 July 2017
Revised: 10 January 2018
Accepted: 10 March 2018
Published: 12 June 2018
##### Authors
 Annette Huber Mathematisches Institut Albert-Ludwigs-Universität Freiburg Freiburg im Breisgau Germany Shane Kelly Department of Mathematics Tokyo Institute of Technology Tokyo Japan