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This paper continues our study of the sheaf associated to K\"ahlerdifferentials in the cdh-topology and its cousins, in positive characteristic,without assuming resolution of singularities. The picture for the sheavesthemselves is now fairly complete. We give a calculation $\mathcal{O}_{cdh}(X)\cong \mathcal{O}(X^{sn})$ in terms of the seminormalisation. We observe thatthe category of representable cdh-sheaves is equivalent to the category ofseminormal varieties. We conclude by proposing some possible connections toBerkovich spaces, and $F$-singularities in the last section. The toolsdeveloped for the case of differential forms also apply in other contexts andshould be of independent interest.

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