''h'' topology

In algebraic geometry, the h topology is a Grothendieck topology introduced by Vladimir Voevodsky to study the homology of schemes. It has several variants, 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] The covering morphisms of the h topology are the universal topological epimorphisms.

The qfh topology has the further restriction that its covering morphisms must be quasi-finite.

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.

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

  1. ↑ SGA I, Exposé IX, définition 2.1
  2. ↑ Suslin and Voevodsky, 4.1

References

  • Suslin, A., and Voevodsky, V., Relative cycles and Chow sheaves, April 1994, .
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.