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

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 ${\mathcal{O}}_{cdh}(X)\cong \mathcal{O}(X^{sn})$ 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.

37 pages

References in corpus (1)

• Algebraic K-theory and descent for blow-ups

Cited by in corpus (1)

• A better comparison of cdh- and ldh-cohomologies