(Čech complex)
For an indexed open cover \(\mcu = \ts{U_i}\) of a topological space \(X\), with an order on its index set, and a sheaf of abelian groups \(\mcf\), put \[ C^p(\mcu, \mcf) = \prod_{i_0 < \cdots < i_p} \mcf(U_{i_0} \intersect \cdots \intersect U_{i_p}) , \] with the alternating-sum differential. \(\check{H}^p(\mcu,\mcf)\) is its cohomology [@Har10a, Chapter III, §4].
(Leray map)
For any open cover \(\mcu\) of \(X\) and sheaf of abelian groups \(\mcf\) there is a natural map \[ \check{H}^p(\mcu, \mcf) \to H^p(X, \mcf), \] the Leray map, induced by comparing the Čech resolution of \(\mcf\) by the sheaves \(\prod j_* \mcf|_{U_{i_0 \cdots i_p}}\) with an injective resolution. It is always an isomorphism in degree \(0\) and injective in degree \(1\), and it is an isomorphism in all degrees when \(H^q(U_{i_0} \cap \cdots \cap U_{i_p}, \mcf) = 0\) for all \(q > 0\) and all finite intersections; see the Čech comparison theorems.
(Leray)
If \(X\) is a noetherian separated scheme, \(\mcu\) is a finite affine open cover, and \(\mcf\) is quasi-coherent, then \[ \check{H}^p(\mcu,\mcf) \cong H^p(X,\mcf) \] for all \(p\) [@Har10a, Theorem III.4.5].
(Affine-cover bound)
If a noetherian separated scheme \(X\) is covered by \(n+1\) affine opens, then \(H^p(X,\mcf)=0\) for \(p>n\) and every quasi-coherent \(\mcf\). The comparison theorem applies because finite intersections of these affine opens are affine; the resulting Čech complex has no terms beyond degree \(n\). In particular the standard cover of \(\PP^n\) gives this bound for projective space [@Har10a, Theorem III.4.5].
(The separatedness hypothesis in the bound)
Glue two copies \(V_1,V_2\) of \(\AA_k^2\) by the identity on their common punctured plane \(W\). The resulting scheme \(X\) is noetherian and has a two-member affine cover. The open Mayer–Vietoris sequence gives \(H^2(X,\OO_X)\cong H^1(W,\OO_W)\), since the positive-degree cohomology of both affine pieces vanishes. This sequence follows by applying sections to a flasque resolution: in each degree the difference map from sections on \(V_1,V_2\) onto sections on \(W\) is surjective by flasqueness. The group on the right is nonzero by P-AGH343PUNCTUREDPLANE. Thus a two-affine cover alone does not imply vanishing in degree two.
An arbitrary open cover need not compute the cohomology, even for a noetherian separated scheme and a quasi-coherent sheaf. For the cover \(\mcu = \{\PP_k^1\}\) of the projective line over a field \(k\) by itself, \(\check{H}^1(\mcu,\OO(-2))=0\) because the Čech complex has a single term, while \(H^1(\PP_k^1,\OO(-2))\cong k\) [@Har10a, Theorem III.5.1].
The affine-cover comparison is a statement in the Zariski topology about quasi-coherent sheaves. On an irreducible topological space, a constant abelian sheaf is flasque: every nonempty open is connected, so its restriction maps are identities or surjections onto zero. Its positive-degree sheaf cohomology therefore vanishes [@Har10a, Proposition III.2.5]. For a complex variety \(X\), singular cohomology is sheaf cohomology of the constant sheaf in the analytic topology, \(H^i_{\operatorname{sing}}(X(\CC),\ZZ)\cong H^i(X^{\operatorname{an}},\underline{\ZZ})\), since \(X^{\operatorname{an}}\) is locally contractible. For \(X=\PP^1_\CC\), the Zariski group \(H^2(X,\underline{\ZZ})\) is \(0\), while \(H^2(X^{\operatorname{an}},\underline{\ZZ})\cong H^2(S^2,\ZZ)\cong\ZZ\).
Dependencies
PR-C9ZEKExactness is checked on stalks, and global sections are only left exactuses