They show that if X is smooth, then , and for various nice non-smooth X, the sheaf recovers objects such as reflexive differentials and torsion-free differentials. Since the Frobenius is an h-covering, in positive characteristic we get for , but analogous results are true if we replace the h-topology with the cdh-topology. By the Nullstellensatz, a morphism of finite presentation towards the spectrum of a field admits a section up to finite extension. That is, there exists a finite field extension and a factorisation . Consequently, for any presheaf and field we have where , resp. , denotes the h-sheafification, resp. etale sheafification. Properties As mentioned above, in positive characteristic, any h-sheaf satisfies . In characteristic zero, we have where is the semi-normalisation (the scheme with the same underlying topological space, but the structure sheaf is replaced with its termwise seminormalisation). Since the h-topology is finer than the Zariski topology, every scheme admits an h-covering by affine schemes. Using abstract blowups and Noetherian induction, if is a field admitting resolution of singularities (e.g., a characteristic zero field) then any scheme of finite type over admits an h-covering by smooth -schemes. More generally, in any situation where de Jong's theorem on alterations is valid we can find h-coverings by regular schemes. Since finite morphisms are h-coverings, algebraic correspondences are finite sums of morphisms. cdh topology The cdh topology on the category of schemes of finite presentation over a qcqs base scheme is generated by: Nisnevich coverings, and families of the form where is a proper morphism of finite presentation, is a closed immersion of finite presentation, and is an isomorphism over . The cd stands for completely decomposed (in the same sense it is used for the Nisnevich topology). As mentioned in the examples section, over a field admitting resolution of singularities, any variety admits a cdh-covering by smooth varieties. This topology is heavily used in the study of Voevodsky motives with integral coefficients (with rational coefficients the h-topology together with de Jong alterations is used). Since the Frobenius is not a cdh-covering, the cdh-topology is also a useful replacement for the h-topology in the study of differentials in positive characteristic. Rather confusingly, there are completely decomposed h-coverings, which are not cdh-coverings, for example the completely decomposed family of flat morphisms . Relation to v-topology and arc-topology The v-topology (or universally subtrusive topology) is equivalent to the h-topology on the category