In algebraic geometry, the h topology is a Grothendieck topology introduced by Vladimir Voevodsky to study the homology of schemes. It combines several good properties possessed by its related "sub"topologies, such as the qfh and cdh topologies. It has subsequently been used by Beilinson to study p-adic Hodge theory, in Bhatt and Scholze's work on projectivity of the affine Grassmanian, Huber and Jörder's study of differential forms, etc. Definition Voevodsky defined the h topology to be the topology associated to finite families of morphisms of finite type such that is a universal topological epimorphism (i.e., a set of points in the target is an open subset if and only if its preimage is open, and any base change also has this property). Voevodsky worked with this topology exclusively on categories of schemes of finite type over a Noetherian base scheme S. Bhatt-Scholze define the h topology on the category of schemes of finite presentation over a qcqs base scheme to be generated by -covers of finite presentation. They show (generalising results of Voevodsky) that the h topology is generated by: fppf-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 . Note that is allowed in an abstract blowup, in which case Z is a nilimmersion of finite presentation. Examples The h-topology is not subcanonical, so representable presheaves are almost never h-sheaves. However, the h-sheafification of representable sheaves are interesting and useful objects; while presheaves of relative cycles are not representable, their associated h-sheaves are representable in the sense that there exists a disjoint union of quasi-projective schemes whose h-sheafifications agree with these h-sheaves of relative cycles. Any h-sheaf in positive characteristic satisfies where we interpret as the colimit over the Frobenii (if the Frobenius is of finite presentation, and if not, use an analogous colimit consisting of morphisms of finite presentation). In fact, (in positive characteristic) the h-sheafification of the structure sheaf is given by . So the structure sheaf "is an h-sheaf on the category of perfect schemes" (although this sentence doesn't really make sense mathematically since morphisms between perfect schemes are almost never of finite presentation). In characteristic zero similar results hold with perfection replaced by semi-normalisation. Huber-Jörder study the h-sheafification of the presheaf of Kähler differentials on categories of schemes of finite type over a characteristic zero base field .