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

Huber-Klawitter, Annette; Kelly, Shane

doi:10.2140/ant.2018.12.649

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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

DOI: 10.2140/ant.2018.12.649

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 $O_{cdh} (X) ≅ O (X^{sn})$ 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.

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

Vol. 12, No. 3, 2018