Let \(X\) be a Noetherian topological space of dimension \(n\). Then \(H^i(X, \mcf) = 0\) for all \(i > n\) and every sheaf of abelian groups \(\mcf\) on \(X\).
Two features distinguish this from every other vanishing theorem on the syllabus. It asks nothing of \(\mcf\): no quasicoherence, no coherence. And the bound is topological, the dimension of the underlying space, not the dimension of a variety over a field.
Noetherianness of the space is load-bearing and is used to run the Noetherian induction on closed subsets; without it the statement fails. Nothing else can be dropped, since \(H^n\) is generally nonzero: \(H^n(\PP^n, \OO(-n-1)) \neq 0\) shows the bound is sharp.
For quasicoherent sheaves there is a cheaper route to the same bound on \(\PP^n\): a cover by \(n+1\) affines makes the Čech complex vanish above degree \(n\). The two arguments give the same number for different reasons, and the Čech one says nothing about non-quasicoherent sheaves.
Dependencies
D-COHDERSheaf cohomology as the derived functor of global sectionsuses