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 Ocdh(X)O(Xsn) 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
Mathematical Subject Classification 2010
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