The Stacks project

59.5 Feats of the étale topology

For a natural number $n \in \mathbf{N} = \{ 1, 2, 3, 4, \ldots \} $ it is true that

More generally, if $X$ is a complex variety, then its étale Betti numbers with coefficients in a finite field agree with the usual Betti numbers of $X(\mathbf{C})$, i.e.,

This is extremely satisfactory. However, these equalities only hold for torsion coefficients, not in general. For integer coefficients, one has

By contrast $H_{Betti}^2(\mathbf{P}^1(\mathbf{C}), \mathbf{Z}) = \mathbf{Z}$ as the topological space $\mathbf{P}^1(\mathbf{C})$ is homeomorphic to a $2$-sphere. There are ways to get back to nontorsion coefficients from torsion ones by a limit procedure which we will come to shortly.