Hypercovering

In mathematics, and in particular homotopy theory, a hypercovering (or hypercover) is a simplicial object that generalises the Čech nerve of a cover. For the Čech nerve of an open cover [math]\displaystyle{ \mathcal U\to X }[/math], one can show that if the space [math]\displaystyle{ X }[/math] is compact and if every intersection of open sets in the cover is contractible, then one can contract these sets and get a simplicial set that is weakly equivalent to [math]\displaystyle{ X }[/math] in a natural way. For the étale topology and other sites, these conditions fail. The idea of a hypercover is to instead of only working with [math]\displaystyle{ n }[/math]-fold intersections of the sets of the given open cover [math]\displaystyle{ \mathcal U }[/math], to allow the pairwise intersections of the sets in [math]\displaystyle{ \mathcal U=\mathcal U_0 }[/math] to be covered by an open cover [math]\displaystyle{ \mathcal U_1 }[/math], and to let the triple intersections of this cover to be covered by yet another open cover [math]\displaystyle{ \mathcal U_2 }[/math], and so on, iteratively. Hypercoverings have a central role in étale homotopy and other areas where homotopy theory is applied to algebraic geometry, such as motivic homotopy theory.

Formal definition

The original definition given for étale cohomology by Jean-Louis Verdier in SGA4, Expose V, Sec. 7, Thm. 7.4.1, to compute sheaf cohomology in arbitrary Grothendieck topologies. For the étale site the definition is the following:

Let [math]\displaystyle{ X }[/math] be a scheme and consider the category of schemes étale over [math]\displaystyle{ X }[/math]. A hypercover is a semisimplicial object [math]\displaystyle{ U_\bullet }[/math] of this category such that [math]\displaystyle{ U_0 \to X }[/math] is an étale cover and such that [math]\displaystyle{ U_{n+1} \to \left(\left(\operatorname{\mathbf{cosk}}_n:= \operatorname{cosk}_n\circ\operatorname{tr}_n\right) U_\bullet\right)_{n+1} }[/math] is an étale cover for every [math]\displaystyle{ n\geq 0 }[/math].

Here, [math]\displaystyle{ U_{n+1} \to \left(\operatorname{\mathbf{cosk}}_n U_\bullet\right)_{n+1} }[/math] is the limit of the diagram which has one copy of [math]\displaystyle{ U_i }[/math] for each [math]\displaystyle{ i }[/math]-dimensional face of the standard [math]\displaystyle{ n+1 }[/math]-simplex (for [math]\displaystyle{ 0 \leq i \leq n }[/math]), one morphism for every inclusion of faces, and the augmentation map [math]\displaystyle{ U_0 \to X }[/math] at the end. The morphisms are given by the boundary maps of the semisimplicial object [math]\displaystyle{ U_\bullet }[/math].

Properties

The Verdier hypercovering theorem states that the abelian sheaf cohomology of an étale sheaf can be computed as a colimit of the cochain cohomologies over all hypercovers.

For a locally Noetherian scheme [math]\displaystyle{ X }[/math], the category [math]\displaystyle{ HR(X) }[/math] of hypercoverings modulo simplicial homotopy is cofiltering, and thus gives a pro-object in the homotopy category of simplicial sets. The geometrical realisation of this is the Artin-Mazur homotopy type. A generalisation of E. Friedlander using bisimplicial hypercoverings of simplicial schemes is called the étale topological type.

References

• Artin, Michael; Mazur, Barry (1969). Etale homotopy. Springer. 
• Friedlander, Eric (1982). Étale homotopy of simplicial schemes. Annals of Mathematics Studies, PUP. 
• Lecture notes by G. Quick "Étale homotopy lecture 2."
• Hypercover in nLab