The Stacks project

59.3 Prologue

These lectures are about another cohomology theory. The first thing to remark is that the Zariski topology is not entirely satisfactory. One of the main reasons that it fails to give the results that we would want is that if $X$ is a complex variety and $\mathcal{F}$ is a constant sheaf then

\[ H^ i(X, \mathcal{F}) = 0, \quad \text{ for all } i > 0. \]

The reason for that is the following. In an irreducible scheme (a variety in particular), any two nonempty open subsets meet, and so the restriction mappings of a constant sheaf are surjective. We say that the sheaf is flasque. In this case, all higher Čech cohomology groups vanish, and so do all higher Zariski cohomology groups. In other words, there are “not enough” open sets in the Zariski topology to detect this higher cohomology.

On the other hand, if $X$ is a smooth projective complex variety, then

\[ H_{Betti}^{2 \dim X}(X (\mathbf{C}), \Lambda ) = \Lambda \quad \text{ for } \Lambda = \mathbf{Z}, \ \mathbf{Z}/n\mathbf{Z}, \]

where $X(\mathbf{C})$ means the set of complex points of $X$. This is a feature that would be nice to replicate in algebraic geometry. In positive characteristic in particular.

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