Sheaf cohomology ($H^p(X,\mathcal{F})$)

Sheaf cohomology consists of the right derived functors of global sections. The zeroth group is $H^0(X,\mathcal{F})=\Gamma (X,\mathcal{F})$, and higher groups measure obstructions to gluing local sections.

Coherent cohomology means sheaf cohomology with coefficients in a coherent sheaf. On a separated Noetherian scheme, a finite affine cover computes it using the Čech cochain complex, because intersections of affine opens are affine and positive-degree quasi-coherent sheaf cohomology vanishes there. Short exact sequences give long exact sequences in sheaf cohomology.

For a coherent sheaf $\mathcal{F}$ on a projective variety over a field, every $H^q(X,\mathcal{F})$ is finite-dimensional, and Grothendieck vanishing gives zero for $q>\dim X$. A closed embedding in projective space and a finite twisting resolution of a coherent sheaf on projective space reduce finiteness to the explicit cohomology of twisting sheaves on projective space. The same resolution proves Serre vanishing after sufficiently positive twisting.

For a cochain complex of sheaves $E$, its $i$th cohomology sheaf is ${\mathcal{H}}^i(E)=\ker (d^i)/\mathrm{im}(d^{i-1})$, where the image and quotient are taken in sheaves. Its stalk at $x$ is the cohomology of the stalk complex, because taking stalks is exact.

A short exact sequence of cochain complexes of sheaves $0\to E\to F\to G\to 0$ induces $\cdots \to {\mathcal{H}}^i(E)\to {\mathcal{H}}^i(F)\to {\mathcal{H}}^i(G)\to {\mathcal{H}}^{i+1}(E)\to \cdots$. Taking stalks is exact, so the construction and exactness follow from the ordinary long exact sequence of cohomology of complexes at every stalk. The stalkwise connecting maps come from the lift-and-differentiate construction and form a sheaf morphism.

For a proper holomorphic map and a coherent sheaf flat over the base, local constancy of the fibre-cohomology dimension makes its higher direct image locally free and gives the displayed base-change identification. Applied to relative holomorphic forms in a smooth proper Kähler family, this constructs the holomorphic cohomology bundles entering its variation of Hodge structure.

Hypercohomology applies the derived global-section functor to a complex of sheaves. One may resolve the complex by an injective double complex, take global sections, and take the cohomology of its total complex. For a complex concentrated in degree zero it is ordinary sheaf cohomology. The relative de Rham complex uses this construction to compute fibre cohomology and its Hodge filtration.

If $0\to \mathcal{F}\to {\mathcal{I}}^0\to {\mathcal{I}}^1\to \cdots$ is an exact resolution of sheaves and each ${\mathcal{I}}^j$ has vanishing positive sheaf cohomology, then the cohomology of its global-section complex computes $H^q(X,\mathcal{F})$. Fine sheaves provide acyclic terms on a paracompact manifold, giving the analytic Dolbeault resolution of holomorphic differential forms.

A positive-degree sheaf cohomology class restricts to zero on some neighbourhood of every point, because a representing cocycle in a flasque resolution is locally a boundary. If a basis closed under finite intersections has vanishing local cohomology in degrees $1,\dots ,i-1$, the neighbourhoods can be chosen in that basis so that the image of the class in $H^i(X,{}_U\mathcal{F})$ is zero. Here ${}_U\mathcal{F}$ is the direct image from an open restriction. Lower local vanishing makes the edge map $H^i(X,{}_U\mathcal{F})\to H^i(U,\mathcal{F}{|}_U)$ injective. Local vanishing of a class alone does not imply global vanishing.

For $X={\mathbb{C}}^2\setminus \{0\}$, the cover $\mathbb{C}\times {\mathbb{C}}^{\ast }$ and ${\mathbb{C}}^{\ast }\times \mathbb{C}$ is acyclic for ${\mathcal{O}}_X$. The first Čech cohomology quotient is represented uniquely by normally convergent doubly negative Laurent series. Under $w_i=1/z_i$ it is $w_1{w}_2\mathcal{O}({\mathbb{C}}_w^2)$, the entire functions divisible by both coordinates. This is an infinite-dimensional analytic vector space; restricting representatives to finite Laurent polynomials loses classes. The class of $1/(z_1{z}_2)$ is nonzero, so $X$ is not a Stein manifold.

A coherent sheaf $\mathcal{F}$ on ${\mathbb{P}}^N$ is $r$-regular if $H^i({\mathbb{P}}^N,\mathcal{F}(r-i))=0$ for every $i>0$. It is then $(r+1)$-regular and $\mathcal{F}(r)$ is globally generated. Persistence follows by induction with a hyperplane avoiding the associated points and its restriction exact sequence. A sheaf on an embedded projective scheme is tested after pushforward to projective space.

An injective resolution is an exact sequence starting with a sheaf and continuing with injective sheaves. It is constructed by successively embedding each quotient in an injective sheaf. Taking global sections and then cohomology computes sheaf cohomology; in particular an injective sheaf has zero higher cohomology.

On a proper scheme over a field, the Euler characteristic of a coherent sheaf is $\chi (X,\mathcal{F})={\sum }_i(-1{)}^i\dim H^i(X,\mathcal{F})$. The sum is finite by Grothendieck vanishing, and long exact sequences in sheaf cohomology make it additive in short exact sequences of sheaves.

For a Cartier divisor $D$ on an $n$-dimensional projective scheme, $\chi (X,{\mathcal{O}}_X(mD))$ is a polynomial in $m$ whose leading term is $D^n{m}^n/n!$. The intersection product uses the fundamental cycle, with the generic multiplicity of each component. Thus the statement also applies to nonreduced schemes.

For an ample line bundle $A$ on a projective scheme and a coherent sheaf $\mathcal{F}$, $H^i(X,\mathcal{F}\otimes A^m)=0$ for every $i>0$ and all sufficiently large $m$.

For a fixed coherent sheaf on a smooth projective curve, one degree bound works for all divisors on an algebraic curve, rather than just powers of one ample divisor. Choose a point $P$. The complement of $P$ is affine, so the surjective transition maps on the finite-dimensional groups $H^1(C,\mathcal{F}(nP))$ eventually kill every class. The Euler characteristic of a coherent sheaf gives $h^0(\mathcal{O}(A))\ge \mathrm{deg}A+1-g$. Every divisor of sufficiently large degree is therefore linearly equivalent to $n_{0}P+E$ with $E$ effective. The point-supported quotient for adding $E$ transfers vanishing to that divisor. Torsion is first removed so that the relevant inclusions are injections of locally free sheaves.

For ample line bundles $L,B$ and a coherent sheaf $\mathcal{F}$ on a projective scheme, one $M$ gives $H^i(X,\mathcal{F}\otimes L^m\otimes B^t)=0$ for every $i>0$, $m\ge M$ and $t\ge 0$. If $B$ is very ample, Serre vanishing for the finitely many fixed twists $\mathcal{F}(-iB)$ makes $\mathcal{F}\otimes L^m$ zero-regular. Persistence of Castelnuovo–Mumford regularity gives all nonnegative $B$-twists. For general ample $B$, use a very ample power and treat finitely many residues of $t$.

Let $H$ be a very ample effective Cartier divisor on a projective scheme $X$, and suppose $L{|}_H$ is ample. Then $H^i(X,L^m)=0$ for $i\ge 2$ and sufficiently large $m$. By uniform Serre vanishing for two ample twists, cohomology of $L^m(tH){|}_H$ vanishes uniformly for $t\ge 0$. The restriction sequence identifies the higher groups of $L^m(tH)$ and $L^m((t+1)H)$ for $i\ge 2$. For fixed $m$, large $t$ kills them by Serre vanishing for $H$; descend to $t=0$. No $H^1$ vanishing is claimed.

Given a coherent sheaf $\mathcal{F}$ and an ample Cartier divisor $A$ on a projective scheme, one integer $a_0$ ensures $H^i(X,\mathcal{F}(aA+N))=0$ for all $i>0$, all $a\ge a_0$, and every nef divisor $N$. The uniformity in $N$ is stronger than Serre vanishing. Fujino's note on Fujita vanishing states the theorem for arbitrary projective schemes over a field.

Let $N$ be a nef Cartier divisor and let the coherent sheaf $\mathcal{F}$ have support dimension $d$. Then $h^i(X,\mathcal{F}(mN))=O(m^{d-i})$ for $1\le i\le d$. This does not require reducedness or smoothness.

Choose a very ample $A$ and fix $a$ sufficiently large for Fujita vanishing. Choose a section of $aA$ avoiding the associated points of $\mathcal{F}$. Its multiplication gives $0\to \mathcal{F}\to \mathcal{F}(aA)\to \mathcal{Q}\to 0$, where $\mathcal{Q}$ has support dimension at most $d-1$. After tensoring with ${\mathcal{O}}_X(mN)$, the middle term has zero higher sheaf cohomology, uniformly in $m$. Hence $h^i(\mathcal{F}(mN))\le h^{i-1}(\mathcal{Q}(mN))$. Induct on support dimension: for $i=1$ use the polynomial bound for sections of a fixed divisor, and for $i>1$ use the induction hypothesis. Degree greater than $d$ vanishes by Grothendieck vanishing.

The top-degree case of cohomology growth for nef twists is bounded independently of $m$. It fills a gap when a dimension induction hypothesis only covers intermediate sheaf cohomology degrees.

For a coherent sheaf on a Noetherian scheme, its sheaf cohomology vanishes in degrees greater than its support dimension. In particular, a zero-dimensional projective scheme has no positive-degree coherent cohomology.

If $0\to \mathcal{F}\to {\mathcal{A}}^0\to {\mathcal{A}}^1\to \cdots$ is an exact sequence of sheaves of abelian groups on $X$ and $H^q(X,{\mathcal{A}}^j)=0$ for every $q>0$ and $j\ge 0$, then sheaf cohomology is computed by the cochain complex of global sections:In particular a flasque resolution computes sheaf cohomology. Acyclicity is required on the space on which cohomology is being computed.

For a continuous map $f:X\to Y$ and a sheaf $\mathcal{F}$ on $X$, the Leray spectral sequence isIf every higher direct image vanishes, its edge maps identify $H^p(X,\mathcal{F})$ with $H^p(Y,f_{\ast }\mathcal{F})$.

A sheaf is flasque when every restriction map is surjective. Flasque sheaves are acyclic for global sections, so $H^i(X,\mathcal{F})=0$ for $i>0$.

For a Noetherian ring $A$, an injective module $I$ gives a flasque sheaf on $\mathrm{Spec}A$. If $U=\mathrm{Spec}A\setminus V(\mathfrak{a})$, clearing denominators on finitely many principal opens identifies $\Gamma (U,\tilde{I})$ with ${\underrightarrow{\lim }}_n{\mathrm{Hom}}_A({\mathfrak{a}}^n,I)$. Each map extends to $A$ by injectivity. Thus every section on $U$ extends globally, and all restriction maps are surjective. This concerns sheafification of an injective module, rather than the different assertion that an injective sheaf of modules is flasque.

For a short exact sequence of sheaves with flasque sheaf kernel, every section of the quotient on any open set lifts to the middle sheaf on that set. Use the Zorn lemma to choose a maximal partial lift, with chains glued by the sheaf gluing axiom. A new local lift differs from the old one on their overlap by a kernel section. Extend that difference by flasqueness, adjust the new lift, and glue, contradicting maximality unless its domain is the entire open set. Consequently a quotient of two flasque sheaves is flasque. With an injective resolution of sheaves, the long exact sequence in sheaf cohomology and repeated dimension shifts prove that flasque sheaves have zero positive-degree cohomology.

A flasque resolution is an exact sequence of sheaves of abelian groups whose terms ${\mathcal{I}}^j$ are flasque sheaves. Every such sheaf embeds by its section germs into the sheaf $U\mapsto {\prod }_{P\in U}{\mathcal{F}}_P$. Its restriction maps are projections, so it is flasque. Apply this construction successively to the cokernels to obtain a resolution. The sheaf cohomology of $\mathcal{F}$ is the cohomology of the cochain complex $\Gamma (X,{\mathcal{I}}^{\bullet })$. A quotient of two flasque sheaves in a short exact sequence of sheaves is flasque: the flasque kernel allows local lifts to be glued into a section of the middle sheaf over the smaller open set; extend that lift using flasqueness of the middle sheaf, and project. Consequently the resolution's successive cokernels are flasque when $\mathcal{F}$ is flasque, and exactness of sections shows that all its positive-degree cohomology vanishes.

Embed a sheaf of abelian groups $\mathcal{F}$ into the sheaf $C(\mathcal{F})(U)={\prod }_{P\in U}{\mathcal{F}}_P$ by all its germs. Restrictions are projections, so $C(\mathcal{F})$ is flasque. Repeat on the successive sheaf cokernels to obtain a canonical flasque resolution. Its complex of global sections computes sheaf cohomology. When $\mathcal{F}$ is itself flasque, the flasque-kernel section-lifting lemma makes every successive quotient flasque and makes the global-section complex exact in positive degrees, proving $H^q(X,\mathcal{F})=0$ for $q>0$ without assuming that vanishing in the construction.

For an open cover $\mathcal{U}=(U_i)$, the Čech cochain group iswith the alternating sum of restrictions as differential. Its cohomology is the Čech cohomology of $\mathcal{F}$ with respect to $\mathcal{U}$.

For $r\ge 1$, a twisting sheaf on projective space has $H^0({\mathbf{P}}_k^r,\mathcal{O}(n))=k[X_0,\dots ,X_r{]}_n$ when $n\ge 0$, and top cohomology with basis the Laurent monomials of total degree $n$ whose exponents are all negative. All intermediate sheaf cohomology vanishes. Their dimensions are $\left(\genfrac{}{}{0ex}{}{n+r}{r}\right)$ and $\left(\genfrac{}{}{0ex}{}{-n-1}{r}\right)$ respectively, with the top group nonzero only for $n\le -r-1$. Decompose the standard Čech cochain complex by exponent vector: no negative exponents gives a simplex with cohomology in degree zero; a nonempty proper negative-support set admits a contracting homotopy inserting a vertex outside that set; all negative exponents give only the top term. For $r=0$, every twist is trivial on $\mathrm{Spec}k$ and has $H^0=k$.

The two affine opens $D(x),D(y)$ cover the punctured affine plane. Their intersection has ring $k[x^{\pm 1},y^{\pm 1}]$. The Čech cohomology in degree one is the quotient by $k[x^{\pm 1},y]+k[x,y^{\pm 1}]$. Its basis consists precisely of the Laurent monomials whose two exponents are negative, so this vector space is infinite-dimensional. Global functions, by contrast, form $k[x,y]$.

On the standard two-affine cover of the projective line, with coordinate $t$, a frame for the twisting sheaf on projective space $\mathcal{O}(n)$ on the second chart equals $t^n$ times the frame on the first. Its Čech cochain complex is $k[t]\oplus k[t^{-1}]\to k[t,t^{-1}]$, with map $(a,b)\mapsto t^{n}b-a$. The kernel has basis $1,t,\dots ,t^n$ for $n\ge 0$. The cokernel has basis $t^{n+1},\dots ,t^{-1}$ for $n\le -2$. The two-chart complex also proves vanishing in degrees at least two.

For analytic holomorphic sections, the two-chart Čech cohomology uses convergent Laurent series, rather than only Laurent polynomials. A frame on the infinity chart equals $z^n$ times a finite-chart frame. Entire finite-chart functions remove Laurent powers $k\ge 0$, and infinity-chart functions remove powers $k\le n$. The positive and negative tails converge to entire chart functions, so only the finite gap $n<k<0$ remains. Thus the first cohomology has basis $z^{n+1},\dots ,z^{-1}$ when $n\le -2$ and is zero when $n\ge -1$. The same transition relation gives global sections of dimension $\max (n+1,0)$.

On $X={\mathbb{A}}_k^1$, let $K=k(t)$ and let $\mathcal{K}$ be its constant sheaf. For the two closed rational points $0,1$, take skyscraper sheaves $S_0,S_1$ with value $K$, and put $\mathcal{F}=\ker (\mathcal{K}\to S_0\oplus S_1)$. The maps on the two stalks are identities, so this is a sheaf surjection. All three displayed outer sheaves are flasque. The induced global map is the diagonal $K\to K^2$, so $H^1(X,\mathcal{F})=K^2/\Delta K\simeq K$. The one-member affine cover has zero positive Čech cochains. This even gives an ${\mathcal{O}}_X$-module example: let regular functions act on all copies of $K$ by multiplication as rational functions, not by evaluation at the closed point. The kernel is not quasi-coherent.

On a good cover, combine Čech degree $p$ and differential-form degree $q$ in the double complex of forms on intersections. The Čech differential and exterior derivative commute, so the total differential $\delta +(-1{)}^{p}d$ squares to zero. A partition of unity and the Poincaré lemma identify its cohomology with both constant-sheaf and de Rham cohomology. The sign convention makes Čech-de Rham curvature descent explicit.

For a finite cover whose nonempty intersections have zero sheaf cohomology in degrees $1,\dots ,i-1$, the displayed sequence is exact for $i>0$. Apply the Čech cochain complex to a flasque resolution. Local primitives and successive primitives of their overlap differences reduce a class with zero restrictions to a Čech cocycle with values in the original sheaf. Vanishing in degree $i$ on all intersections is not needed.

The Čech cochain complex places sections on $(p+1)$-fold intersections in degree $p$ and uses the alternating sum of restriction maps as its differential.

The Čech differential sends a cochain on $(p+1)$-fold intersections to the alternating sum of its restrictions to $(p+2)$-fold intersections. Each pair of omitted indices occurs twice with opposite signs in the composite, so ${\delta }^2=0$. It defines the Čech cohomology of the cover.

For a finite affine cover of a semi-separated scheme, every finite intersection is affine. The augmented sheaf Čech cochain complex is exact: near each point, an open of the cover contains the neighbourhood under consideration, and inserting its index contracts the complex. Its terms are acyclic for a quasi-coherent sheaf, by affine cohomology vanishing and the affine-intersection property. Alternatively, apply the section-level Čech cochain complex to a flasque resolution; the two computations of the resulting double complex identify Čech cohomology with sheaf cohomology. Stalks of direct images from opens outside those opens need not vanish, so exactness must not be justified by treating these direct images as extensions by zero.

For a finite family generating the unit ideal in a commutative ring, this augmented Čech cochain complex of any module is exact. One proof clears the finitely many denominators and cocycle relations by powers $f_i^N$, writes $1={\sum }_i{a}_i{f}_i^N$, and contracts an alternating cocycle by the weighted insertions ${\sum }_i{a}_i{f}_i^N{c}_{iI}$. The argument uses vanishing criterion in a module localization and does not require the module to be finitely generated.

The degree-$p$ Čech cochain group for an ordered cover is ${\prod }_{i_0<\cdots <i_p}\mathcal{F}(U_{i_0}\cap \cdots \cap U_{i_p})$.

A Čech cochain is a cocycle when its alternating coboundary vanishes. For a multiplicative one-cochain $(g_{ij})$, this says $g_{ij}{g}_{jk}=g_{ik}$ on triple intersections.

A Čech coboundary is the image of a cochain in the preceding degree. Multiplicatively, a zero-cochain $(h_i)$ changes a one-cocycle by $g_{ij}\mapsto h_i^{-1}{g}_{ij}{h}_j$.

If every nonempty finite intersection of members of an open cover has vanishing higher sheaf cohomology for $\mathcal{F}$, then the cover's Čech cohomology computes $H^p(X,\mathcal{F})$. An affine open cover of a separated scheme satisfies this condition for a quasi-coherent sheaf because its finite intersections are affine.

If $i:Z\hookrightarrow X$ is a closed immersion of Noetherian schemes, $X$ is separated, and $\mathcal{F}$ is quasi-coherent on $Z$, thenIndeed, an affine cover of $X$ induces an affine cover of $Z$, and the two associated Čech cochain complexes are termwise identical.

If a separated scheme has an acyclic affine cover by $r$ open sets, then every quasi-coherent sheaf on it has vanishing sheaf cohomology in degrees at least $r$. The corresponding Čech cochain complex has no terms in those degrees.

A sheaf of modules over the sheaf of smooth functions is fine when every locally finite open cover admits a subordinate family of sheaf endomorphisms summing to the identity. Fine sheaves on a paracompact space are acyclic, so their positive-degree sheaf cohomology vanishes.

For $X=U\cup V$, the short exact sequence of sheaves obtained by restricting to $U$, $V$, and $U\cap V$ induces a long exact sequence

A short exact sequence of sheaves $0\to {\mathcal{F}}^{\prime }\to \mathcal{F}\to {\mathcal{F}}^{\prime \prime }\to 0$ induces a natural long exact sequence

For a smooth projective $n$-dimensional variety $X$ over a field and a coherent sheaf $\mathcal{F}$, Serre duality gives a perfect pairing between $H^q(X,\mathcal{F})$ and ${\mathrm{Ext}}^{n-q}(\mathcal{F},{\omega }_X)$, where ${\omega }_X$ is the canonical sheaf.

For vector bundles on a smooth projective curve, a dualizing sheaf is a line bundle $D_C$ with a trace $H^1(C,D_C)\to k$ making the evaluation pairings $H^i(C,E)\times H^{1-i}(C,E^{\vee }\otimes D_C)\to k$ perfect for $i=0,1$. Choose a finite generically separable map to the projective line. The trace-dual module for this finite map, tensored with ${\Omega }_{{\mathbb{P}}^1}^1$, is ${\Omega }_C^1$ by the different–differential identity. Finite pushforward and residue duality on the projective line then prove the pairings, rather than assuming the curve-duality theorem to establish them.

Use the rational principal-parts resolution on an algebraic curve to represent a class by finitely many rational differential principal parts. Sum their algebraic residues of a rational differential. The sum of residues of a global rational differential is zero: trace to the projective line, where partial fractions prove the assertion. Hence the sum descends to cohomology. Pairing an invertible sheaf with its differential-valued dual gives the perfect Serre duality pairing. Since the global regular functions on an integral projective curve over an algebraically closed field are $k$, evaluation at the constant function $1$ identifies this trace canonically with an isomorphism to $k$.

The coefficient of $t^{-1}dt$ gives a trace on $H^1({\mathbb{P}}^1,{\Omega }^1)$. Multiplication followed by this trace pairs the Laurent bases of the Čech cohomology of twists on the projective line perfectly: $t^a$ pairs with $t^{-a-1}$. For any vector bundle, evaluation into ${\Omega }^1$ defines the same natural pairing. Saturate a rational rank-one subspace to a line subbundle, and induct on rank using the long exact sequence in sheaf cohomology and compatibility of evaluation with connecting maps. This proves the duality without using the later splitting theorem.

On ${\mathbb{P}}_k^n$ with $n>0$, let $\mathcal{F}=i_{\ast }k$ at a rational point. It has one-dimensional global sections, but its dual of a sheaf is zero: a local map from the residue field into the local polynomial ring must have image annihilated by a nonzero parameter, which is impossible in an integral domain. Thus $H^n({\mathcal{F}}^{\vee }(-n-1))=0$. Serre duality for arbitrary coherent sheaves instead uses an Ext functor; replacing this by the ordinary dual can lose torsion information.

For a compact complex $n$-manifold and a holomorphic vector bundle $E$, analytic Serre duality givesThe pairing integrates the wedge product of bundle-valued Dolbeault cohomology representatives, contracting $E$ with its dual.