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Write $U_0=D_{+}(t_0)=\mathrm{Spec}\mathbb{C}[x]$, where $x=t_1/t_0$, and $U_1=D_{+}(t_1)=\mathrm{Spec}\mathbb{C}[y]$, with $y=1/x$ on the overlap. Since $\mathcal{G}$ is a quasi-coherent sheaf, it is associated to a $\mathbb{C}[x]$-module $M$ on $U_0$. The preimages under the open inclusion $f$ of $U_0,U_1,U_0\cap U_1$ are $U_0,D(x),D(x)$, all affine schemes. Thus $f$ is an affine morphism and its direct image sheaf $f_{\ast }\mathcal{G}$ is quasi-coherent by direct image of a quasi-coherent sheaf under an affine morphism.

The two-chart cover is an acyclic cover by vanishing of quasi-coherent cohomology on an affine scheme, so the acyclic cover theorem identifies its Čech cohomology with sheaf cohomology. Its only potentially nontrivial positive-degree differential isThis is surjective because of the second summand. There are no normalized Čech terms in degree two or higher for a two-open cover, soThis is the acyclic direct image from an affine chart of the projective line. The pushforward need not itself be a flasque sheaf; the acyclic affine cover is what justifies this computation.

Choose an ordering of the index set $I$. The degree-$q$ Čech cochain group isFor $s=(s_{i_0\cdots i_q})$, the Čech coboundary is the alternating sum of restrictionsThe identity ${\delta }^2=0$ makes this the Čech cochain complex, and the required Čech cohomology is

For an open cover $\mathcal{U}=(U_i)$ and a sheaf $\mathcal{F}$, the Čech cochain complex iswith the alternating sum of restriction maps as differential. Its cohomology is the Čech cohomology ${\check{H}}^p(\mathcal{U},\mathcal{F})$.

Cover ${\mathbb{P}}_k^1$ by $U_0=D_{+}(x_0)$ and $U_1=D_{+}(x_1)$, and write $t=x_1/x_0$. This is an acyclic affine cover for the twisting sheaf on projective space $\mathcal{O}(-1)$. In compatible trivializations, its degree-one Čech quotient isThe first summand contains every nonnegative power of $t$ and the second every negative power, so

For an indexed open cover $\mathcal{U}=(U_i{)}_{i\in I}$ and a sheaf $\mathcal{F}$, the Čech cohomology cochain groups areThe differential is the alternating sum of restrictions:Since $d^2=0$, the cohomologyis well defined.

Write $A=k[x,y]$ and $\mathfrak{m}=(x,y)$. Since $f(0,0)\ne 0$, the principal open subscheme $D(f)=\mathrm{Spec}{A}_f$ contains $\mathfrak{m}$, andCover $V$ by the two affine opens $D(xf)$ and $D(yf)$, whose intersection is $D(xyf)$. The degree-zero part of the resulting Čech cohomology complex givesinside the fraction field of $A$. The last equality follows because $A_f$ is a unique factorization domain and a rational function regular after localizing at both $x$ and $y$ has no possible prime factor left in its denominator.

The same affine cover is acyclic, so its degree-one Čech group computes sheaf cohomology and givesBefore localizing at $f$, the quotienthas the $k$-basisWriting $f=c+h$ with $c=f(0,0)\in k^{\times }$ and $h\in (x,y)$, multiplication by $h$ is locally nilpotent on $Q$: for each negative monomial, a sufficiently high power of $(x,y)$ moves every term into $A_x+A_y$. Hence $c+h$ acts invertibly on $Q$ by a finite geometric series on each element. Localizing at $f$ therefore leaves $Q$ unchanged, andThe displayed infinite basis proves that this vector space is infinite-dimensional.

For an oriented manifold $M$ of dimension $n$ and a compact locally contractible subset $A$, relative duality givesFor a general compact subset the right side is interpreted as Čech cohomology; local contractibility permits the ordinary singular cohomology used above. The orientation fixes the isomorphism, and its naturality identifies restriction of an ambient fundamental class with restriction of the constant class in degree zero.

The local duality isomorphism is first obtained on coordinate balls. Excision and the Mayer–Vietoris sequence make the identifications compatible under unions; passage to neighbourhood limits yields the compact-subset formulation. On regular compact neighbourhoods the same construction is Poincare-Lefschetz duality followed by the neighbourhood limit. This explains both the relative homology pair and the cohomological degree shift.

In particular, if $M$ is connected, closed and oriented and $A$ is an embedded closed ball, the relative group is $\mathbb{Z}$ only in degree $n$. The map $H_n(M)\to H_n(M,M\setminus A)$ is an isomorphism, since it corresponds to $H^0(M)\to H^0(A)$. The long exact sequence of a pair therefore gives $H_k(M\setminus A)\cong H_k(M)$ for $k<n$ and zero top homology of the complement. This application concerns removal of the closed ball itself.

For a topological space $X$ and abelian group $A$, singular cohomology is the cohomology of the cochain complex $C^n(X;A)=\mathrm{Hom}(C_n(X;\mathbb{Z}),A)$, where $C_n$ is the singular chain group. Its differential is dual to the singular boundary. This theory has relative cohomology, long exact sequences and the Excision theorem. On spaces that are not locally contractible it can differ from Čech cohomology.

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.