In algebraic geometry, the h topology is a Grothendieck topology introduced by Vladimir Voevodsky to study the homology of schemes.
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.
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[3][4]).
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
They show (generalising results of Voevodsky) that the h topology is generated by: Note that
is allowed in an abstract blowup, in which case Z is a nilimmersion of finite presentation.
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.
over the Frobenii (if the Frobenius is of finite presentation, and if not, use an analogous colimit consisting of morphisms of finite presentation).
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.
of Kähler differentials on categories of schemes of finite type over a characteristic zero base field
, but analogous results are true if we replace the h-topology with the cdh-topology.
As mentioned above, in positive characteristic, any h-sheaf satisfies
is the semi-normalisation (the scheme with the same underlying topological space, but the structure sheaf is replaced with its termwise seminormalisation).
More generally, in any situation where de Jong's theorem on alterations is valid we can find h-coverings by regular schemes.
is generated by: It is the universal topology with a "good" theory of compact supports.
[6] 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
The v-topology (or universally subtrusive topology) is equivalent to the h-topology on the category
of all qcqs schemes, neither of the v- nor the h- topologies are finer than the other:
is the non-open, non-closed prime Rydh (2010, Example 4.3)).
However, we could define an h-analogue of the fpqc topology by saying that an hqc-covering is a family
Indeed, any subtrusive morphism is submersive (this is an easy exercise using Rydh (2010, Cor.1.5 and Def.2.2)).
is a v-cover if and only if it is an arc-cover (for the statement in this form see Bhatt & Mathew (2018, Prop.2.6)).
That is, in the Noetherian setting everything said above for the v-topology is valid for the arc-topology.