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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.

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 $k[x,y]$.

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 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.

The Dolbeault theorem identifies this group with first Čech cohomology of the trivial holomorphic line bundle. On the two standard affine charts, a holomorphic overlap function has a Laurent series. Its nonnegative powers extend to the chart at zero, and its negative powers extend to the chart at infinity. Every overlap cocycle is therefore a coboundary. Consequently the Dolbeault operator from functions to $(0,1)$-forms is surjective.

For $X={\mathbb{P}}^1\times \mathbb{C}$, the two standard affine charts and their intersection form an acyclic cover for the sheaves of holomorphic forms. Their Čech cohomology gives $H_{\bar{\partial }}^{0,0}=\mathcal{O}(\mathbb{C})$ and $H_{\bar{\partial }}^{2,1}=\mathcal{O}(\mathbb{C})$, with all other groups for $p\in \{0,2\}$ zero. For the top-form sheaf, the transition $d(1/z)=-z^{-2}dz$ removes every Laurent series power except $z^{-1}$ from the first cohomology quotient; its coefficient may be an arbitrary entire function on the affine factor.

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.

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 $\max (n+1,0)$.

For $n\ge 1$, the top Čech cohomology of the standard affine cover of projective space is the degree-$m$ quotient of the Laurent ring by the sum of rings in which at least one variable has not been inverted. Its basis consists of Laurent monomials $X_0^{a_0}\cdots X_n^{a_n}$ with every $a_i<0$ and ${\sum }_i{a}_i=m$. Hence it vanishes for $m\ge -n$ and has dimension $\left(\genfrac{}{}{0ex}{}{-m-1}{n}\right)$ for $m\le -n-1$.

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.

A line bundle trivial on each open set is determined by unit-valued transition functions on pairwise intersections. Their cocycle identities make the local trivial bundles glue; changing frames changes the cocycle by a coboundary. The tensor product of sheaves multiplies the transition functions, giving the displayed group isomorphism with Čech cohomology.

For a well-order on the indices of an open cover, the ordered Čech cochain complex uses only strictly increasing tuples, with the alternating restriction Čech differential. It computes the same Čech cohomology as the complex using all tuples. If every finite intersection is nonempty and the coefficient constant sheaf has the same abelian group $A$ on each intersection, inserting the least index $\alpha$ gives a cochain homotopy: set $(hs{)}_I=s_{\alpha ,I}$ when $\alpha$ is not in $I$, and $(hs{)}_I=0$ otherwise. Expanding the alternating sums gives $\delta h+h\delta =\mathrm{id}$ in every positive degree, since the terms deleting an index of $I$ cancel and the term deleting $\alpha$ is $s_I$. Hence positive Čech cohomology vanishes in this situation; degree zero remains $A$.

Use homogeneous coordinates $x_0,\dots ,x_r$ on projective space and letThese $r+1$ open subschemes cover $X$. Each is a closed subscheme of the affine scheme $D_{+}(x_i)$, hence is an affine scheme.

Every nonempty finite intersection is affine as well. Indeed, inside $D_{+}(x_{i_0})$ the intersection with $D_{+}(x_{i_1}),\dots ,D_{+}(x_{i_p})$ is the principal open subscheme obtained by invertingIntersecting it with the closed subscheme $X$ gives $U_{i_0}\cap \cdots \cap U_{i_p}$, again an affine scheme.

For any quasi-coherent sheaf $F$, vanishing of quasi-coherent cohomology on an affine scheme makes every such intersection acyclic for $F$. The acyclic cover theorem therefore identifies the sheaf cohomology of $F$ with the Čech cohomology of this cover. Its alternating Čech cochain complex has termswhich vanish for $p>r$: one cannot choose $p+1$ distinct indices from $r+1$ indices. ConsequentlyThe choice $N=\max \{1,r\}$ meets the strict requirement $N>0$ even when $r=0$. It is independent of $F$, and applies to all quasi-coherent sheaves, including those which are not coherent sheaves. This is the cohomological dimension bound from an affine cover.

Discard empty members of the open cover: every Čech cochain group factor supported on an empty intersection is the zero abelian group, so this changes no Čech cohomology. Every remaining member is the one-point space $X$; the index set is nonempty even if the same open subset is listed repeatedly. Choose a well-order with least index $\alpha$. Use the ordered Čech cochain complex permitted in the question:All restriction maps are identities. For $q\ge 1$ define a cochain homotopy $h_q:C^q\to C^{q-1}$ byWe verify $\delta h_q+h_{q+1}\delta ={\mathrm{id}}_{C^q}$ for every $q\ge 1$. Fix $I=(i_0<\cdots <i_q)$. If $i_0=\alpha$, then $(h_{q+1}\delta s{)}_I=0$. In $(\delta h_{q}s{)}_I$, the term deleting $i_0$ is $s_I$ and every other term is zero because its retained first index is $\alpha$. Thus the identity holds on $I$.

If $i_0>\alpha$, thenAdding proves the identity in this case also. Each formula uses only finitely many terms on each component, so it remains valid for an infinite open cover and arbitrary abelian group $A$.

If $s\in C^q$ is a Čech cocycle, then $\delta s=0$, and the identity gives $s=\delta (h_{q}s)$: every Čech cocycle in positive degree is a Čech coboundary. Therefore, using the permitted comparison with ordered Čech cohomology,The identity is asserted in positive degrees; degree zero has $H^0\cong A$, so an unaugmented contraction to zero in every degree would be incorrect.

In an irreducible topological space, any two nonempty open subsets intersect. Let $W\subseteq X$ be a nonempty open subset. Two nonempty subsets open in $W$ are also open in $X$, and hence intersect; thus $W$ is itself an irreducible topological space. If $W$ were disconnected, a separation into two disjoint nonempty open subsets would contradict this intersection property. Hence $W$ is a connected space. The empty open subset is connected as well under the usual convention.

Discard empty members of $\mathcal{U}$ as in (b). Every finite intersection $U_{i_0}\cap \cdots \cap U_{i_q}$ is nonempty by repeated use of the intersection property and is an open subset, hence a connected space. A locally constant function on this intersection is constant. It follows that the constant sheaf hasand every restriction map between these nonempty intersections is the identity on $A$. The whole Čech cochain complex is therefore the same as for the indexed open cover of a point having the same index set. The cochain homotopy constructed in (b) applies component by component and provesThe argument concerns the constant sheaf of locally constant functions; arbitrary coefficient sheaves on an irreducible topological space need not have the same Čech cohomology.

Suppose, to the contrary, that $H^1(X,\mathcal{F})=0$ for every quasi-coherent sheaf $\mathcal{F}$. Since $X$ is a quasi-compact scheme, part (a) makes it an affine scheme.

Every affine scheme $X=\mathrm{Spec}A$ is a separated scheme: its diagonal morphism over $\mathrm{Spec}\mathbb{Z}$ is induced by the surjective ring homomorphism $A{\otimes }_{\mathbb{Z}}A\to A$, $a\otimes b\mapsto ab$, so it is a closed immersion. This contradicts the assumption that $X$ is not a separated scheme. ThereforeIn fact, the version of the Serre criterion for affineness using ideals shows that $\mathcal{F}$ may be chosen to be a quasi-coherent sheaf of ideals. This argument does not use a comparison with Čech cohomology on the nonseparated scheme.

Suppose $X=U_1\cup \cdots \cup U_n$ with each $U_i$ an affine open subscheme. Every affine scheme is a quasi-compact topological space: a cover by principal open subsets $D(f_{\lambda })$ means that the $f_{\lambda }$ generate the unit ideal; an expression $1={\sum }_{j=1}^r{a}_j{f}_{{\lambda }_j}$ gives a finite subcover. Every open cover has a refinement by principal open subsets, so this proves the general covering property.

Restrict an arbitrary open cover of $X$ to each $U_i$ and choose a finite subcover there. The union of these finitely many finite families covers $X$. Thus $X$ is a quasi-compact scheme.

Now assume $X$ is a separated scheme. By (c), every finite intersection of the $U_i$ is an affine scheme, and hence has vanishing positive sheaf cohomology for the structure sheaf. This affine open cover therefore computes sheaf cohomology using its Čech cochain complex. Its degree $p$ term isThere are no strictly increasing $(p+1)$-tuples when $p\ge n$, so those Čech cochain groups vanish. ConsequentlyThe same argument works for every quasi-coherent sheaf. The relevant comparison requires the acyclicity of the finite intersections; the bare assertion that arbitrary Čech cohomology equals sheaf cohomology would be too broad.

Interpretation of the removed locus. Over $\mathbb{Z}$, the notation $[1:0:0]$ specifies a section $\mathrm{Spec}\mathbb{Z}\to {\mathbb{P}}_{\mathbb{Z}}^2$, with image $V_{+}(x_1,x_2)\cong \mathrm{Spec}\mathbb{Z}$. We remove this entire closed subscheme. This is the interpretation forced by the asserted open cover $X=U_1\cup U_2$. Removing one closed point in a particular fiber would not give that open cover. The native PDF uses the same notation as the transcription.

Put $a=x_0/x_1$, $b=x_2/x_1$ on $U_1$. On $U_2$ put $c=x_0/x_2$, $d=x_1/x_2$. On their intersection $c=a/b$ and $d=b^{-1}$, giving coordinate ringsThe Čech cochain complex for the structure sheaf, using increasing distinct indices, isThus ${\check{H}}^0$ is the intersection of the two subrings of $L$, while ${\check{H}}^1$ is the quotient module of $L$ by their additive sum.

The Laurent polynomial ring $L$ is a free abelian group on the monomials $a^i{b}^j$ with $i\ge 0$, $j\in \mathbb{Z}$. The first subring has the basis $j\ge 0$. The second has the basisso its allowed exponents satisfy $j\le -i$. A monomial belongs to both precisely when $i=j=0$. The monomials belonging to neither satisfy $i\ge 2$ and $-i<j<0$. Since the subgroups are spanned by subsets of the same basis, their sum introduces no other relations. ThereforeHere the square brackets denote cohomology classes, and the direct sums are as abelian groups. In particular, $[a^2{b}^{-1}]=[x_0^2/(x_1{x}_2)]$ is a nonzero cohomology class. This scheme is a separated scheme, being an open subscheme of projective space over $\mathbb{Z}$, and is a quasi-compact scheme by (d). Both charts and their intersection are affine schemes, so this Čech cohomology equals sheaf cohomology. The nonzero first sheaf cohomology of the quasi-coherent structure sheaf rules out an affine scheme:

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.