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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 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.

For a quasi-compact separated scheme and a global regular function $f$, global sections on $X_f$ are the localization of global sections by $f$. Choose a finite affine open cover. Its intersections are affine, so the sheaf gluing axiom describes global sections as the kernel of the difference map between two finite products of affine coordinate rings. Exactness of localization preserves this kernel, and localization commutes with those finite products. Localizing the same cover gives the claimed isomorphism. This applies before knowing whether $X$ itself is affine.

Let $\mathcal{U}=(U_0,\dots ,U_m)$ be a finite affine open cover of the projective scheme $X$. For a coherent sheaf $\mathcal{F}$, form the Čech cochain complexThe two ways to omit any pair of indices have opposite signs, giving ${\delta }^2=0$. Its cohomology groups areA projective scheme over a field is a separated scheme. Finite intersections of its affine opens are affine, and a coherent sheaf is quasi-coherent. Higher sheaf cohomology of a quasi-coherent sheaf on an affine scheme vanishes. Thus this is an acyclic cover and the acyclic cover theorem identifies the displayed Čech cohomology with $H^p(X,\mathcal{F})$. In particular its degree-zero kernel glues compatible local sections to $\Gamma (X,\mathcal{F})$. A cover by $m+1$ opens also gives vanishing for $p>m$.

For the cohomology of twists on projective space, take $S=k[X_0,\dots ,X_r]$ with the usual grading and $U_i=D_{+}(X_i)$. On an intersection indexed by the nonempty set $J$,This follows directly from the construction of the twisting sheaf on projective space: sections of the twist are the degree-$n$ homogeneous fractions in the corresponding localization. The Čech differential preserves each Laurent monomial $X^a=X_0^{a_0}\cdots X_r^{a_r}$, where $a_i\in \mathbb{Z}$ and ${\sum }_i{a}_i=n$. Such a monomial appears precisely in summands withThe complex therefore decomposes as the direct sum of finite-dimensional cochain complexes indexed by these exponent vectors. Their differentials have only the alternating signs of the simplex incidence maps.

If $N(a)=\varnothing$, this is the ordinary unaugmented simplex cochain complex: its degree-zero kernel consists of the common value on every vertex and has dimension one, while all higher cohomology vanishes. If $\varnothing \ne N(a)\ne \{0,\dots ,r\}$, choose $v\notin N(a)$. A cochain homotopy inserting $v$ into the ordered index list, with the sign of that insertion, contracts this subcomplex. Insertion or deletion of $v$ preserves the condition $N(a)\subseteq J$; in $\delta h+h\delta$, terms inserting and deleting different vertices cancel in pairs, while the term inserting then deleting $v$ is the identity. Thus this entire monomial subcomplex is acyclic. Finally, if all $a_i<0$, the monomial occurs only in the full intersection, in degree $r$, and contributes one copy of $k$ there.

For $r\ge 1$, these three cases give the complete answer:In the first case counting nonnegative exponent vectors gives $h^0=\left(\genfrac{}{}{0ex}{}{n+r}{r}\right)$. In the top case write $a_i=-1-b_i$ with $b_i\ge 0$; then $\sum b_i=-n-r-1$, givingThis is the Laurent-monomial description of top cohomology on projective space. All intermediate degrees vanish for every twist, and the top degree vanishes when $n\ge -r$. If $r=0$, then ${\mathbf{P}}_k^0=\mathrm{Spec}k$ and every twist is trivial, so $H^0({\mathbf{P}}_k^0,\mathcal{O}(n))=k$ for every integer $n$, with all higher groups zero.

Choose an ordered affine open cover $\mathcal{U}=(U_i)$ of the separated scheme. For a quasi-coherent sheaf $\mathcal{F}$, the Čech cochain groups and differential areTerms corresponding to empty intersections are zero. Restriction commutes with restriction, so the terms in $d^2$ cancel in pairs. DefineHere an affine cover computes the Čech groups independently of the chosen affine cover. The reason is that finite intersections of affine opens in a separated scheme are affine: the intersection of two is the inverse image of the closed diagonal inside their affine product. Repeating gives the finite-intersection assertion. Quasi-coherent sheaves have no higher cohomology on these intersections. Apply an injective resolution of the sheaf and form the double complex of its Čech cochains. In one direction the higher cohomology vanishes on every affine intersection, leaving precisely the displayed Čech complex. In the other direction the augmented Čech complex of the flasque resolution terms is exact, leaving the global-section resolution computing sheaf cohomology. Thus the two cohomologies agree, and refinement maps between affine-cover complexes induce the same groups. No such affine-intersection claim is being made for arbitrary nonseparated schemes.

For an exact sequence of quasi-coherent sheaves, sections on an affine intersection give an exact sequence of modules, including surjectivity: this follows from the equivalence between quasi-coherent sheaves and modules on an affine scheme. Taking products yields a short exact sequence of Čech cochain complexes. To describe the connecting map, lift a degree-$p$ cocycle in $C^p({\mathcal{F}}^{\prime \prime })$ to a cochain in $C^p(\mathcal{F})$. Its differential maps to zero in $C^{p+1}({\mathcal{F}}^{\prime \prime })$ and therefore is a cocycle in $C^{p+1}({\mathcal{F}}^{\prime })$. Changing the lift adds a coboundary there; changing the original representative does the same. This defines the connecting homomorphism. The same lift-and-differentiate argument shows that a class is in its kernel exactly when it lifts to a cocycle in the middle complex, and verifies exactness at the other terms. Thus the resulting sequence isThe sequence must be in quasi-coherent sheaves for this affine-sections argument; that is the intended category here.

For the punctured affine plane, use $U=D(x)$ and $V=D(y)$. Their rings are $k[x^{\pm 1},y]$ and $k[x,y^{\pm 1}]$, and the intersection ring is $k[x^{\pm 1},y^{\pm 1}]$. The two-open Čech complex isConsequentlyThe denominator contains precisely the span of Laurent monomials for which at least one exponent is nonnegative. Every Laurent polynomial has a unique monomial expansion, so the remaining monomials are linearly independent in the quotient and span it. Thereforea countably infinite-dimensional vector space. The brackets here denote cohomology classes. This calculates the Čech cohomology of the punctured affine plane, rather than merely observing that some nonzero cohomology exists.

Take the projective line with a doubled point over $\mathbb{C}$, obtained by gluing two copies of ${\mathbb{P}}_{\mathbb{C}}^1$ along the complement of one point. After every base change of a morphism of schemes, a closed subset has closed image from each of the two projective-line charts, so its total image, the union of those two images, is closed. The structure morphism is therefore universally closed. The two doubled points have no disjoint neighborhoods, so the scheme is not separated and hence is not proper.

The coincidence locus of two scheme morphisms is the fibre productThe diagonal morphism ${\Delta }_Y$ is a locally closed immersion, and this property is stable under base change of a morphism of schemes, so $E\hookrightarrow X$ is locally closed. The universal property of a fibre product says that a morphism $T\to X$ factors through $E$ exactly when $f{|}_T=g{|}_T$. Thus $E$ is the largest locally closed subscheme on which they coincide. If $Y$ is separated, ${\Delta }_Y$ is a closed immersion, and its base change $E\hookrightarrow X$ is closed.

Choose a finite affine cover $\mathcal{U}=(U_j)$ of the Noetherian scheme $X$. Because $X$ is separated, every finite intersection $U_J$ is affine. Its inverse image $V_J=Z\cap U_J$ under the closed immersion is also affine. By the definition of the direct image sheaf,The Čech complexes for $i_{\ast }\mathcal{F}$ on $\mathcal{U}$ and for $\mathcal{F}$ on the induced cover $(Z\cap U_j)$ are therefore identical, including their restriction maps. Both affine covers are acyclic for the relevant quasi-coherent sheaves, so the acyclic cover theorem givesfor every $q$. This is cohomology under a closed immersion.

For $X={\mathbb{P}}_k^n$, use its $n+1$ standard affine opens. The induced cover of $Z$ is still acyclic, and its Čech complex has no cochains in degrees greater than $n$. The cohomological dimension bound from an affine cover therefore yields

A group scheme over $k$ is a $k$-scheme $G$ equipped with multiplication $m:G{\times }_{k}G\to G$, identity $e:\mathrm{Spec}k\to G$, and inversion $j:G\to G$ satisfying the associativity, identity, and inverse diagrams.

ConsiderThe diagonal morphism is ${\Delta }_{G/k}=q^{-1}(e)$. For a finite-type $k$-scheme the rational identity point is closed, so its inverse image is closed. Thus the diagonal is a closed immersion and every such group scheme over a field is a separated scheme.

In characteristic $p>0$, the infinitesimal additive groupis a nonreduced group scheme. Its comultiplication is $t\mapsto t\otimes 1+1\otimes t$, which is well defined because $(t\otimes 1+1\otimes t{)}^p=t^p\otimes 1+1\otimes t^p$.

Every quasi-coherent sheaf on an affine scheme has zero higher sheaf cohomology. Together with the equivalence between modules and quasi-coherent sheaves on an affine scheme, this makes affine varieties the basic acyclic pieces for computing cohomology. An affine open cover of a separated scheme has affine finite intersections, so the acyclic cover theorem identifies its Čech cohomology with sheaf cohomology.