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.

Definition

Define a morphism of schemes to be submersive or a topological epimorphism if it is surjective on points and its codomain has the quotient topology, i.e., a subset of the codomain is open if and only if its preimage is open. A morphism is universally submersive or a universal topological epimorphism if it remains a topological epimorphism after any base change.[1][2]

Voevodsky defines the h topology on the category of schemes to be the topology associated to finite families \( {\displaystyle \{p_{i}:U_{i}\to X\}} \) of morphisms of finite type such that \( {\displaystyle \amalg U_{i}\to X} \) is a universal topological epimorphism.

The qfh topology is associated to families as above, with the further restriction that each p i {\displaystyle p_{i}} p_{i} must be quasi-finite.

cdh topology

While defined on all schemes, the *h* and *qfh* topology are only ever used on Noetherian schemes. The *h* topology has various non-equivalent extensions to non-Noetherian schemes including the *ph* topology^{[3]} and the *v* topology.

The proper *cdh* topology is defined as follows. Let *p* : *Y* → *X* be a proper morphism. Suppose that there exists a closed immersion *e* : *A* → *X*. If the morphism *p*^{−1}(*X* − *e*(*A*)) → *X* − *e*(*A*) is an isomorphism, then *p* is a covering morphism for the *cdh* topology. The *cd* stands for *completely decomposed* (in the same sense it is used for the Nisnevich topology). An equivalent definition of a covering morphism is that it is a proper morphism *p* such that for any point *x* of the codomain, the fiber *p*^{−1}(*x*) contains a point rational over the residue field of *x*.

The *cdh* topology is the smallest Grothendieck topology whose covering morphisms include those of the proper *cdh* topology and those of the Nisnevich topology.

Properties

The h topology combines a number of useful properties of its various "sub"topologies. Since if is finer than the Zariski topology, h-locally every scheme is affine. Since it is finer than the Nisnevich_topology, h-locally regular immersions look like zero sections of vector bundles. It is also finer than the étale topology and the fppf topology.

In a different direction, it is finer than the qfh topology, so h locally, algebraic correspondences are finite sums of morphisms.[4] Finally, every proper surjective morphism is an h covering, so in any situation where de Jong's theorem on alterations is valid, h locally all schemes are regular.

Relation to v-topology

The v-topology (or universally subtrusive topology) is equivalent to the h-topology on Noetherian schemes. On more general schemes, the v-topology has more covers.

Notes

SGA I, Exposé IX, définition 2.1

Suslin and Voevodsky, 4.1

A cohomological bound for the h-topology

Suslin, Voevodsky, Singular homology of abstract algebraic varieties

References

Suslin, A., and Voevodsky, V., Relative cycles and Chow sheaves, April 1994, [1].

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