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An open cover is acyclic for a specified sheaf if its restriction to every nonempty finite intersection has zero higher sheaf cohomology. The acyclic cover theorem then identifies the cover's Čech cohomology with sheaf cohomology. On a separated scheme, an affine open cover is acyclic for every quasi-coherent sheaf.

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.

Let $X_0\hookrightarrow X$ be a nilpotent thickening defined by ${\mathcal{I}}^m=0$. If $X$ is an affine scheme, its closed subscheme $X_0$ is an affine scheme. Conversely, suppose $X_0$ is an affine scheme. The underlying topological spaces are identical, so $X$ is a quasi-compact scheme. For any quasi-coherent sheaf $\mathcal{F}$, each subquotient ${\mathcal{I}}^j\mathcal{F}/{\mathcal{I}}^{j+1}\mathcal{F}$ is naturally a quasi-coherent sheaf on $X_0$. By sheaf cohomology under a closed inclusion and vanishing of quasi-coherent cohomology on an affine scheme, its first sheaf cohomology on $X$ vanishes. The long exact sequence in sheaf cohomology and descending mathematical induction through this finite filtration prove $H^1(X,\mathcal{F})=0$. The Serre criterion for affineness now makes $X$ an affine scheme.

Let $P$ have an affine open neighbourhood $U$, and let $Y=X\setminus U$. Evaluation gives ${\mathcal{I}}_Y\twoheadrightarrow k_P$ with kernel ${\mathcal{I}}_Y\cap {\mathcal{I}}_P$. If first sheaf cohomology of every coherent ideal sheaf vanishes, a global section $f$ of ${\mathcal{I}}_Y$ can be chosen with $f(P)=1$. Then $X_f\subseteq U$ is a principal open subset of an affine variety and is affine.

On a Stein manifold $X$, every coherent analytic sheaf $\mathcal{F}$ has $H^q(X,\mathcal{F})=0$ for $q>0$. In particular this applies to the sheaf of holomorphic functions. If every nonempty finite intersection in an open cover is Stein, these vanishing results and the acyclic cover theorem compute the corresponding sheaf cohomology from its Čech cochain complex.

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.

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.

On a complex manifold, a sheaf of modules over the sheaf of holomorphic functions is coherent analytic if it is locally finitely generated and the kernel of every morphism ${\mathcal{O}}^r\to \mathcal{F}$ is locally finitely generated. The structure sheaf and locally free finite-rank sheaves are examples. Cartan theorem B gives vanishing of their higher sheaf cohomology on a Stein manifold.

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.

A Noetherian scheme is affine exactly when every coherent ideal sheaf has zero first sheaf cohomology. One direction is vanishing of quasi-coherent cohomology on an affine scheme. For the converse, ideal-sheaf vanishing produces affine principal neighbourhoods from ideal-sheaf vanishing. A finite such cover yields a unit-ideal certificate from a principal affine cover, and those affine charts glue to the spectrum of a commutative ring of global sections. The criterion, including the finite-type ideal version for quasi-compact quasi-separated schemes, is recorded in Stacks Project, Section 30.3.

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.

In a proper flat family of algebraic curves, if the first sheaf cohomology of a line bundle vanishes on every fibre of an open set, its direct image there is a vector bundle of rank given by the Riemann-Roch theorem and commutes with arbitrary base change. Locally a finite free complex computes both fibre sheaf cohomology groups. Vanishing of the last cokernel makes its last map surjective; splitting that map leaves a finite locally free kernel in degree zero. This is the mechanism behind charts of the Picard scheme of a curve and evaluation descriptions of the theta divisor.

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.

For a commutative ring $R$, remove the section $[1:0:0]$ from ${\mathbb{P}}_R^2$. The two remaining standard charts have coordinate rings $R[a,b]$ and $R[a/b,b^{-1}]$ inside the Laurent polynomial ring $R[a,b,b^{-1}]$. Their intersection consists of constants: their monomial exponent regions are $i\ge 0,j\ge 0$ and $i\ge 0,j\le -i$, respectively. The Čech cohomology in degree one is free over $R$ on the missing monomials $a^i{b}^{-k}$ with $1\le k<i$. Equivalently its basis consists of the classes of $x_0^{m+n}/(x_1^m{x}_2^n)$ for $m,n\ge 1$. Degree zero is $R$ and every degree at least two is zero. The charts and their intersection are affine schemes, so this computes sheaf cohomology. If $R$ is nonzero, the first sheaf cohomology is nonzero and this scheme is not an affine scheme.

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 $H^0=k$.

For an affine morphism $f:X\to Y$ and a quasi-coherent sheaf $\mathcal{F}$ on $X$, the direct image sheaf has the same sheaf cohomology on $Y$ as $\mathcal{F}$ on $X$. To prove this, take a flasque resolution $\mathcal{F}\to {\mathcal{I}}^{\bullet }$. Its direct images are flasque sheaves. On each affine open $V\subset Y$, the complex of sections of the direct images is $\Gamma (f^{-1}V,{\mathcal{I}}^{\bullet })$. By vanishing of quasi-coherent cohomology on an affine scheme, this augmented complex is exact. Checking on the affine basis proves that $f_{\ast }{\mathcal{I}}^{\bullet }$ resolves $f_{\ast }\mathcal{F}$. Its global sections are identical to those of the original resolution.

For a line bundle $M$ on $C\times T$, with $C$ a smooth projective curve, this function is locally constant. Twist both $M$ and $M^{-1}$ by a large power of an ample line bundle on $C$. Near a chosen fiber their first sheaf cohomology groups vanish. The Riemann-Roch theorem and upper semicontinuity of their zeroth sheaf cohomology groups then bound the degree from above and below by the degree on that fiber. Connectedness of $T$ is necessary for a single constant across all fibers.

The complex of smooth differential forms resolves the constant complex sheaf by the Poincaré lemma and consists of fine sheaves. Its global cohomology therefore agrees with sheaf cohomology of that constant sheaf. A good-cover Čech-de Rham double complex gives an explicit comparison, used in Čech-de Rham curvature descent.

From a short exact sequence of sheaves $0\to K_{j+1}\to G_j\to K_j\to 0$, the long exact sequence in sheaf cohomology gives the displayed isomorphism whenever $H^q(X,G_j)=H^{q+1}(X,G_j)=0$. Repeating it along a locally free resolution shifts a fixed group to successively higher degrees. If all relevant kernel sheaves are quasi-coherent sheaves and there is a uniform vanishing bound for their sheaf cohomology, sufficiently many shifts force the original group to vanish. The cohomological dimension bound from an affine cover supplies such a bound for a closed subscheme of projective space.

Étale cohomology is sheaf cohomology on the site whose objects are étale morphisms $Y\to X$ and whose coverings are jointly surjective families of such morphisms. It provides cohomological invariants for algebraic varieties over fields of positive characteristic. In Deligne-Lusztig theory, compactly supported groups with $\ell$-adic coefficients, for $\ell$ different from the field characteristic, give finite-dimensional representations of the finite groups acting on the varieties.