Let \(X = \Spec A\) with \(A\) Noetherian and \(\mcf \in \QCoh(X)\). Then \(H^0(X, \mcf) = \globsec{X;\mcf}\) and \(H^i(X, \mcf) = 0\) for all \(i > 0\).
(The proof in one line)
Put \(M = \globsec{X;\mcf}\), so \(\mcf \cong \tilde M\). Take an injective resolution \(M \injects I^\bullet\) in \(\amod\) and sheafify: each \(\tilde{I^j}\) is flasque, hence acyclic, so \(\tilde{I^\bullet}\) computes the cohomology. Applying \(\globsec{X;\wait}\) returns the original exact complex \(M \injects I^\bullet\), and its higher cohomology is zero.
(Čech computation)
Let \(\mcf\) be a quasicoherent sheaf on a scheme \(X\).
If \(X\) is affine and \(\mathfrak{U} = \{U_1, \ldots, U_r\}\) is a finite cover of \(X\) by affine opens whose finite intersections are affine, the augmented Čech complex \(0 \to \mcf(X) \to C^0(\mathfrak{U}, \mcf) \to C^1(\mathfrak{U}, \mcf) \to \cdots\) is exact; so \(\check{H}^i(\mathfrak{U}, \mcf) = 0\) for \(i > 0\).
If \(X\) is quasicompact and separated, any two finite affine covers \(\mathfrak{U}\), \(\mathfrak{V}\) give the same Čech cohomology: \(\check{H}^i(\mathfrak{U}, \mcf) \cong \check{H}^i(\mathfrak{V}, \mcf)\); for \(X\) Noetherian both equal \(H^i(X, \mcf)\) [@Har10a, Theorem III.4.5].
(Hypotheses)
Quasicoherence is the load-bearing hypothesis, and it is where the whole argument lives: the theorem is false for general sheaves of abelian groups, which see the topology of \(X\) rather than the module \(M\). Noetherianness is only used to know that \(\tilde I\) is flasque for \(I\) injective; the statement is true for any affine scheme, but that version is harder.
This is the direction of Serre’s criterion that is a theorem about affines, and the other direction is the criterion proper. It is also the reason every computation is done on an affine cover: on the pieces of the cover there is nothing above degree \(0\), so all the cohomology comes from the gluing, which is exactly what the Čech complex records.
Dependencies
D-COHFLQFlasque sheaves and acyclicityuses