Sheaf cohomology \(H^i(X,-)\) is the \(i\)th right derived functor of \(\Gamma(X,-)\) on sheaves of abelian groups. For a quasi-coherent sheaf on a noetherian separated scheme, the Čech complex of a finite affine open cover computes it. The cover of \(\PP^n_A\) by the \(n+1\) opens \(D_+(x_i)\) gives \(H^*(\PP^n_A,\OO(d))\) for every \(d\).
Čech cohomology and the comparison with sheaf cohomology (Tag D-PTIW0)
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 \(H^2(X^{\operatorname{an}},\underline{\ZZ})\cong H^2(S^2,\ZZ)\cong\ZZ\).
The cohomology of $\OO_{\PP^n}(d)$ (Tag T-IJW1K)
Let \(A\) be a ring, let \(n\ge1\), put \(S=A[x_0,\ldots,x_n]\), and write \(\PP^n=\PP_A^n\). Then
\(H^0(\PP^n, \OO(d)) = S_d\), a free \(A\)-module of rank \(\binom{n+d}{n}\) for \(d \geq 0\), and \(0\) for \(d < 0\);
\(H^n(\PP^n,\OO(d))\) is the free \(A\)-module on the monomials \(x_0^{a_0}\cdots x_n^{a_n}\) with every \(a_i\le-1\) and \(\sum_i a_i=d\). It is zero for \(d>-n-1\) and has rank \(\binom{-d-1}{n}\) for \(d\le-n-1\). It is \(A\)-linearly dual to \(S_{-d-n-1}\);
\(H^p(\PP^n, \OO(d)) = 0\) for \(0 < p < n\) and for \(p > n\), for every \(d\).
[@Har10a, Theorem III.5.1] The projective-space cohomology calculation gives these formulas over every ring \(A\).
(Projective dimension zero)
For \(n=0\), the scheme \(\PP_A^0\) is \(\Spec A\), and every \(\OO(d)\) is trivial. Thus \(H^0(\PP_A^0,\OO(d))\cong A\) for every integer \(d\), and all positive-degree cohomology groups vanish. The negative-degree sections are represented on its single standard chart by \(Ax_0^d\), as in P-AGH2510SATIDEAL, step <2>1.
(Consequences over a field)
For \(n\ge2\), \(H^1(\PP_k^n,\OO(d))=0\) for every integer \(d\). On \(\PP_k^1\), one has \[ h^0(\OO(d))=\max\{d+1,0\},\qquad h^1(\OO(d))=\max\{-d-1,0\}=h^0(\OO(-d-2)). \] In particular \(h^1(\OO(-2))=1\). The Euler sequence gives \(\Omega_{\PP_k^1/k}^1\cong\OO(-2)\), so the projective line has no nonzero global regular differentials. The duality between the top and bottom groups is Serre duality, with canonical sheaf \(\omega_{\PP_k^n}=\OO(-n-1)\).
The shift \(-n-1\) in the duality between \(H^n(\PP^n,\OO(d))\) and \(S_{-d-n-1}\), where \(S=A[x_0,\ldots,x_n]\), is the degree of the canonical sheaf \(\omega_{\PP^n}=\OO(-n-1)\) of differentials. The perfect pairing \(H^0(\PP^n,\OO(d))\times H^n(\PP^n,\OO(-d-n-1))\to H^n(\PP^n,\OO(-n-1))\cong A\) is Serre duality on \(\PP^n\).
\(H^1\): lifting sections and line bundles
$H^1$ classifies line bundles and extensions (Tag PR-ET5PQ)
\(\Pic(X) \cong H^1(X, \OO_X^*)\), and for \(\OO_X\)-modules \(\mcf, \mcg\) the extensions \[ 0 \to \mcf \to \mce \to \mcg \to 0 \] up to equivalence are classified by \(\Ext^1(\mcg,\mcf)\), which is \(H^1(X, \mcf \tensor \dualof{\mcg})\) when \(\mcg\) is locally free.
The isomorphism is the Čech description read backwards: a line bundle is trivial on some cover, the transition functions \(g_{ij} \in \OO^*(U_i \intersect U_j)\) satisfy the cocycle condition, and changing the trivialisations changes them by a coboundary. So \(H^1\) of the units is the group of line bundles, with tensor product as the group law.
This is the same phenomenon as the failure of right exactness on sections in stalks and exactness, and it is worth saying so: in both cases \(H^1\) measures the failure of local data to be global, once as sections that do not glue and once as trivialisations that do not agree.
On a complex manifold the exponential sequence makes it explicit, \[ H^1(X,\OO) \to H^1(X,\OO^*) \to H^2(X,\ZZ) , \] where the second map is the first Chern class: the topological type of a line bundle, with \(H^1(X,\OO)\) measuring how many holomorphic structures carry it.
For an exact sequence \(0\to\mcf'\to\mcf\to\mcf''\to0\) of sheaves of abelian groups, a section \(s\in\Gamma(X,\mcf'')\) lifts to \(\Gamma(X,\mcf)\) if and only if \(\delta(s)=0\) in \(H^1(X,\mcf')\). On a cover \(\{U_i\}\) with local lifts \(t_i\in\mcf(U_i)\) of \(s\), the Čech cocycle \((t_j-t_i)|_{U_i\cap U_j}\) with values in \(\mcf'\) represents \(\delta(s)\). Under \(\Pic X\cong H^1(X,\OO_X^\times)\), the Čech cocycle of a line bundle is its family of transition functions \(g_{ij}\in\OO_X^\times(U_i\cap U_j)\).
Dévissage
Dévissage of coherent sheaves (Tag D-COHDEVISSAGE)
Let \(X\) be a Noetherian scheme and \(\mcf\) a coherent \(\OO_X\)-module. There is a filtration \(0 = \mcf_0 \subseteq \mcf_1 \subseteq \cdots \subseteq \mcf_m = \mcf\) by coherent subsheaves such that for each \(j\) there are an integral closed subscheme \(i_j \colon Z_j \hookrightarrow X\) and a nonzero coherent ideal sheaf \(\mathcal{I}_j \subseteq \OO_{Z_j}\) with \(\mcf_j / \mcf_{j-1} \cong i_{j*} \mathcal{I}_j\).
(Dévissage)
Dévissage is the reduction of a statement about all coherent sheaves on a Noetherian scheme to the sheaves \(i_* \mathcal{I}\) of the proposition. If a property \(P\) of coherent sheaves holds for an extension whenever it holds for the subsheaf and the quotient, and holds for \(i_* \mathcal{I}\) for every integral closed subscheme \(i \colon Z \hookrightarrow X\) and nonzero coherent ideal \(\mathcal{I} \subseteq \OO_Z\), then \(P\) holds for every coherent sheaf, by induction along the filtration.
Grothendieck’s proof that \(R^i f_* \mcf\) is coherent for \(f\) proper and \(\mcf\) coherent uses dévissage together with Noetherian induction on the support: it reduces to sheaves on an integral scheme, where Chow’s lemma reduces the proper case to the projective case.