BREAKING NEWS
Hypercovering

## Summary

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 ${\displaystyle {\mathcal {U}}\to X}$, one can show that if the space ${\displaystyle X}$ 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 ${\displaystyle X}$ 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 ${\displaystyle n}$-fold intersections of the sets of the given open cover ${\displaystyle {\mathcal {U}}}$, to allow the pairwise intersections of the sets in ${\displaystyle {\mathcal {U}}={\mathcal {U}}_{0}}$ to be covered by an open cover ${\displaystyle {\mathcal {U}}_{1}}$, and to let the triple intersections of this cover to be covered by yet another open cover ${\displaystyle {\mathcal {U}}_{2}}$, 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 ${\displaystyle X}$  be a scheme and consider the category of schemes étale over ${\displaystyle X}$ . A hypercover is a simplicial object ${\displaystyle U_{\bullet }}$  of this category such that ${\displaystyle U_{0}\to X}$  is an étale cover and such that ${\displaystyle U_{n+1}\to (\operatorname {cosk} _{n}\operatorname {sk} _{n}U_{\bullet })_{n+1}}$  is an étale cover for every ${\displaystyle n\geq 0}$ .

Here, ${\displaystyle (\operatorname {cosk} _{n}\operatorname {sk} _{n}U_{\bullet })_{n+1}}$  is the limit of the diagram which has one copy of ${\displaystyle U_{i}}$  for each ${\displaystyle i}$ -dimensional face of the standard ${\displaystyle n+1}$ -simplex (for ${\displaystyle 0\leq i\leq n}$ ), and one morphism for every inclusion of faces. The morphisms are given by the boundary maps of the simplicial object ${\displaystyle U_{\bullet }}$ .

## 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 ${\displaystyle X}$ , the category ${\displaystyle HR(X)}$  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