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 Grassmannian, Huber and Jörder's study of differential forms, etc.
Definition Voevodsky defined the h topology to be the topology associated to finite families { p i : U i → X } {\displaystyle \{p_{i}:U_{i}\to X\}} of morphisms of finite type such that ⨿ U i → X {\displaystyle \amalg U_{i}\to X} 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 S c h / S f t {\displaystyle Sch_{/S}^{ft}} of schemes of finite type over a Noetherian base scheme S. Bhatt-Scholze define the h topology on the category S c h / S f p {\displaystyle Sch_{/S}^{fp}} of schemes of finite presentation over a qcqs base scheme S {\displaystyle S} to be generated by v {\displaystyle v} -covers of finite presentation. They show (generalising results of Voevodsky) that the h topology is generated by:
fppf-coverings, and families of the form { X ′ → X , Z → X } {\displaystyle \{X'\to X,Z\to X\}} where
X ′ → X {\displaystyle X'\to X} is a proper morphism of finite presentation,
Z → X {\displaystyle Z\to X} is a closed immersion of finite presentation, and
X ′ → X {\displaystyle X'\to X} is an isomorphism over X ∖ Z {\displaystyle X\setminus Z} . Note that X ′ = ∅ {\displaystyle X'=\varnothing } is allowed in an abstract blowup, in which case Z is a nilimmersion of finite presentation.
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