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First consider the preliminary cohomological facts. The exam's term flabby means flasque sheaf: every restriction map is surjective. For open sets $V\subseteq U$, lift a section of ${\mathcal{F}}_3(V)$ to ${\mathcal{F}}_2(V)$ using the given surjectivity when ${\mathcal{F}}_1$ is flasque. Extend that lift to $U$ using the flasque sheaf ${\mathcal{F}}_2$, then project to ${\mathcal{F}}_3(U)$. This proves that ${\mathcal{F}}_3$ is a flasque sheaf.

One construction of a flasque resolution starts with the sheaf ${\mathcal{C}}^0(\mathcal{F})(U)={\prod }_{P\in U}{\mathcal{F}}_P$, with pointwise restriction. It is a sheaf of abelian groups with surjective restrictions, and sending a section to all its germs embeds $\mathcal{F}$ in it. Repeating on the cokernels constructs an exact sequencewith each ${\mathcal{I}}^j$ flasque. Define sheaf cohomology by the cohomology of the cochain complex of global sections of such a resolution; the groups are independent of the chosen flasque resolution. If $\mathcal{F}$ is itself flasque, the quotient result just proved makes every successive cokernel flasque. Applying the assumed surjectivity of sections to each successive short exact sequence of sheaves proves exactness of the global-section complex in positive degrees. Thus $H^i(X,\mathcal{F})=0$ for $i>0$.

Now let $K=k(X)$ be the function field of the smooth algebraic curve. Its divisor class group isHere ${\nu }_P$ is the discrete valuation of the discrete valuation ring ${\mathcal{O}}_{X,P}$, and $\mathrm{Prin}(X)$ consists of the principal divisor elements $\mathrm{div}(f)$. These sums have finite support: on a finite affine cover, represent $f$ as a fraction of regular functions; each nonzero numerator or denominator has only finitely many zeros on a curve, since a proper closed subset of a Noetherian one-dimensional irreducible space is finite.

For a divisor on an algebraic curve $D={\sum }_P{n}_{P}P$, define the line bundle associated to a divisor byIn this expression the valuation condition applies to nonzero $f$. If $t_P$ is a uniformizer, the stalk is $t_P^{-n_P}{\mathcal{O}}_{X,P}$. Shrinking around $P$ removes all other zeros and poles of $t_P$ and all other points in the support of $D$, so this also gives an actual local generator. Thus ${\mathcal{O}}_X(D)$ is a locally free sheaf of rank one. Multiplication gives an isomorphismon every stalk. If $D=\mathrm{div}(g)$, then ${\mathcal{O}}_X(D)=g^{-1}{\mathcal{O}}_X$, so the construction factors through a homomorphism $\mathrm{Cl}(X)\to \mathrm{Pic}(X)$.

It is injective: if ${\mathcal{O}}_X(D)$ is trivial, an isomorphism from ${\mathcal{O}}_X$ supplies a nonzero global rational generator $h$. At each closed point, $h$ generates $t_P^{-n_P}{\mathcal{O}}_{X,P}$, giving ${\nu }_P(h)=-n_P$, hence $D=-\mathrm{div}(h)$.

It is surjective: take a line bundle $\mathcal{L}$ and a nonzero rational section $s$, obtained by choosing a frame on any nonempty trivializing open set. If $e_i$ is a local frame, write $s=f_i{e}_i$ with $f_i\in K^{\ast }$. Since changes of frame are units, ${\nu }_P(f_i)$ is independent of the frame near $P$. A finite trivializing cover shows that these numbers have finite support. Set $D={\sum }_P{\nu }_P(f_i)P$. The map $f\mapsto fs$ gives ${\mathcal{O}}_X(D)\cong \mathcal{L}$, because at $P$ the condition ${\nu }_P(f)+{\nu }_P(f_i)\ge 0$ is exactly regularity of $fs$. Replacing $s$ by $gs$ changes $D$ by $\mathrm{div}(g)$, so this construction is well defined on classes. This proves the divisor class group and Picard group of a smooth curve identification

The sheaf of nonzero rational functions on an irreducible variety ${\mathcal{K}}_X^{\ast }$ is a flasque sheaf. Indeed, its sections on any nonempty open subset are $K^{\ast }$: irreducibility makes nonempty open subsets connected and makes any two of them meet, so locally constant rational-function values must agree. Restrictions between such opens are identities; restriction to the empty set is also surjective. The preliminary argument therefore givesApply the long exact sequence in sheaf cohomology to the short exact sequence of sheaves of multiplicative abelian groupsIts relevant part isPart (b) identifies the first map with the principal divisor map $K^{\ast }\to \mathrm{Div}(X)$. Exactness therefore identifies its cokernel with $H^1(X,{\mathcal{O}}_X^{\ast })$. Combining this with part (a) givesThe connecting map sends a divisor to the class of the unit ratios of its local rational representatives; these are precisely the transition data of the associated line bundle, up to the usual choice of inverse transition convention.

Choose a flasque resolution of the sheaf of abelian groups $\mathcal{F}$,A flasque sheaf has surjective restriction maps; such resolutions exist for every sheaf of abelian groups. Apply the global section functor to obtain a cochain complex. The sheaf cohomology groups areFor $i=0$, the denominator is zero and $H^0(X,\mathcal{F})=\Gamma (X,\mathcal{F})$. Different resolutions give canonically isomorphic groups; the resolution principle for sheaf cohomology is what permits other acyclic resolutions to compute these same groups.

The restriction maps of this point sheaf satisfy the presheaf composition identities: every possible restriction is either the identity on $\mathbb{C}$ or the unique map to the zero group. We check the sheaf identity and gluing axioms for any open cover $U={\bigcup }_i{U}_i$.

If $p\notin U$, every section group in the cover is zero, so a compatible family has the unique zero gluing. If $p\in U$, at least one $U_i$ contains $p$. For any two members containing $p$, their intersection also contains $p$, and compatibility forces their complex values to be equal, since the restriction maps there are identities. Call the common value $c$. Members not containing $p$ have only the zero section. The section $c\in {\mathbb{C}}_p(U)$ restricts to every member of the family, proving existence of a gluing; its restriction to any member containing $p$ determines $c$, proving uniqueness.

This also proves local identity for sections, and ${\mathbb{C}}_p(\varnothing )=0$ handles the empty open set. HenceIt is the point-pushforward sheaf, usually called a skyscraper sheaf. Its restrictions are all surjective, so it is also a flasque sheaf. No separation assumption on $X$ is needed for the preceding sheaf-axiom argument.

The kernel of the restriction mapconsists exactly of sections whose germs vanish outside $Z$, namely ${\Gamma }_Z(X,\mathcal{F})$. This proves exactness. If $\mathcal{F}$ is a flasque sheaf, the restriction map is surjective by definition.

Now let $0\to {\mathcal{F}}_1\to {\mathcal{F}}_2\to {\mathcal{F}}_3\to 0$ be exact. The global section functor is left exact, so a section of ${\Gamma }_Z(X,{\mathcal{F}}_2)$ mapping to zero lifts uniquely to a global section of ${\mathcal{F}}_1$. Its germs outside $Z$ vanish because ${\mathcal{F}}_1\to {\mathcal{F}}_2$ is injective at each stalk of a sheaf. It therefore lies in ${\Gamma }_Z(X,{\mathcal{F}}_1)$, proving exactness of the supported-section sequence.

Suppose in addition that ${\mathcal{F}}_1$ is flasque and take $s_3\in {\Gamma }_Z(X,{\mathcal{F}}_3)$. Surjectivity on global sections gives a lift $s_2\in \Gamma (X,{\mathcal{F}}_2)$. On $X\setminus Z$, its restriction comes from some $t_1\in \Gamma (X\setminus Z,{\mathcal{F}}_1)$ by left exactness. Extend $t_1$ to ${\tilde{t}}_1\in \Gamma (X,{\mathcal{F}}_1)$ using flasqueness. Then $s_2-{\tilde{t}}_1$ maps to $s_3$ and vanishes outside $Z$, proving surjectivity on the right.

For a closed inclusion $i:X\hookrightarrow Y$, direct image sheaf formation is exact: stalks on $X$ are unchanged and stalks outside $X$ are zero. It also preserves flasque sheaves. Pushing forward a flasque resolution thus gives a resolution computing cohomology on $Y$, and its complex of global sections is identical to the original one on $X$. This proves the displayed isomorphism in every nonnegative degree.

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 an irreducible algebraic variety $X$ with function field $K$, this sheaf of abelian groups assigns $K^{\ast }$ to every nonempty open set, with identity restriction maps, and assigns the trivial group to the empty set. Irreducibility ensures that any two nonempty open subsets meet, so these sections satisfy the sheaf gluing axiom. It is a flasque sheaf and has zero higher sheaf cohomology.