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