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Let $V\subseteq U$ be open subsets, and let $s\in {\mathcal{F}}_3(V)$. The assumed flasque-kernel section-lifting lemma lifts $s$ to $\tilde{s}\in {\mathcal{F}}_2(V)$. Since ${\mathcal{F}}_2$ is a flabby sheaf, extend $\tilde{s}$ to a section on $U$, then project to ${\mathcal{F}}_3(U)$. Its restriction is $s$. Thus every restriction map of ${\mathcal{F}}_3$ is surjective, proving that the quotient of two flabby sheaves in a short exact sequence is flabby.

For the construction of sheaf cohomology, embed a sheaf of abelian groups $\mathcal{F}$ into the sheafby sending a section to all its germs. This is injective because a section with every germ zero is zero. Arbitrary families of stalk elements glue point by point, and restriction maps are projections; hence $C(\mathcal{F})$ is a flabby sheaf. Set ${\mathcal{Q}}^0=\mathcal{F}$, ${\mathcal{I}}^j=C({\mathcal{Q}}^j)$ and ${\mathcal{Q}}^{j+1}={\mathcal{I}}^j/{\mathcal{Q}}^j$. Compose the quotient map with the next embedding to obtain the flasque resolutionThe sheaf cohomology groups are the cohomology groups of the resulting cochain complex of global sections:The usual comparison of resolutions identifies this construction with the right derived functors of global sections; the construction above is the Godement resolution.

If $\mathcal{F}$ is itself a flabby sheaf, the quotient argument shows inductively that every ${\mathcal{Q}}^j$ is flabby. The assumed lifting result therefore makesan exact sequence for every $j$. The global-section complex is consequently exact in every positive degree. In particular,This establishes the vanishing from the section-lifting property rather than assuming it in advance.

For the smooth algebraic curve $X$, a divisor on an algebraic curve is a finite formal sum $D={\sum }_P{n}_{P}P$ of closed points with integer coefficients. At each closed point, the local ring of a smooth algebraic curve is a discrete valuation ring; write its normalized discrete valuation as $v_P$. For $f\in k(X{)}^{\ast }$ the principal divisor is $\mathrm{div}(f)={\sum }_P{v}_P(f)P$. Only finitely many coefficients are nonzero: zeros and poles are proper closed subsets in the Noetherian Zariski topology of a curve. The divisor class group is the abelian groupEquivalently, its elements are divisors on an algebraic curve modulo linear equivalence of divisors.

For a sheaf of abelian groups $\mathcal{F}$ on a topological space $X$, putThese arbitrary families of germs form a sheaf, and restrictions are projections. Extending a family by zero proves that $C(\mathcal{F})$ is flasque. Sending a section to all its germs gives an injective sheaf map $\mathcal{F}\to C(\mathcal{F})$, since a section with every germ zero vanishes locally and hence globally.

Set $Q^{-1}=\mathcal{F}$, $I^j=C(Q^{j-1})$ and $Q^j=\mathrm{coker}(Q^{j-1}\to I^j)$, using the sheaf cokernel. Compose $I^j\to Q^j\to I^{j+1}$ to obtain the Godement resolutionIt is exact by construction, with flasque terms. Define sheaf cohomology by the cohomology of its global-section complex:The resolution principle for sheaf cohomology identifies this with the computation using any flasque resolution.

The elementary input for acyclicity is the flasque-kernel section-lifting lemma: in a short exact sequence with flasque kernel, sections of the quotient lift over every open subset. Consequently a quotient of two flasque sheaves is flasque, because a quotient section lifts and that lift extends. If $\mathcal{F}$ itself is flasque, induction through $0\to Q^{j-1}\to I^j\to Q^j\to 0$ makes every $Q^j$ flasque. The same lifting lemma makes these sequences exact on global sections. Hence the global-section complex has no positive-degree cohomology, provingThis deduction uses the lifting lemma, not the desired cohomological vanishing as an assumption.

For the rational-section construction, add two representative sections after restricting to the intersection of their dense domains. Multiply by a representative rational function in the same way. Finite intersections of dense open sets are dense, and further restricting both representatives does not change their resulting class. The module axioms hold on their common domain. Thus rational sections form a module over $\mathrm{Rat}(X)$; irreducibility is not needed for this module statement. When $X$ is irreducible, $\mathrm{Rat}(X)$ is the function field $K=k(X)$.

Suppose now that $\mathcal{F}$ is locally free and $X$ irreducible. A germ at $P$ is represented on a nonempty open neighbourhood, which is dense, so it defines a rational section. Two representatives of the germ agree on a neighbourhood of $P$ and therefore give the same class. If the class is zero, trivialize $\mathcal{F}$ on a neighbourhood of $P$. Its coefficient functions vanish on a dense open subset; regular functions on a reduced irreducible variety with this property vanish identically. The germ is therefore zero. This proves the stalk inclusion into rational sections of a locally free sheaf:Local freeness matters: a nonzero torsion germ can vanish on a dense open set.

On an irreducible algebraic curve, every proper closed subset is finite, and every open subset is quasi-compact because the Zariski space is Noetherian. Let $R=\mathrm{Rat}(\mathcal{F})$. The constant sheaf $\mathcal{R}(\mathcal{F})$ has sections $R$ on every nonempty open set: such opens are irreducible and hence connected. DefineRestriction drops the components outside the smaller open set. For a compatible family over an open cover of $U$, take a finite subcover by quasi-compactness. The union of the finite supports of these finitely many sections is finite. Their agreeing components therefore glue to a unique finite-support tuple on $U$. This verifies the sheaf gluing axiom, including arbitrary covers, rather than merely asserting that a presheaf direct sum is always a sheaf.

A rational section is regular on some dense open set, whose complement on a curve is finite. Its classes $[s{]}_P\in R/{\mathcal{F}}_P$ therefore have finite support, defining a sheaf map $\mathcal{R}(\mathcal{F})\to \mathcal{P}(\mathcal{F})$. At $P$, the first sheaf has stalk $R$. The second has stalk $R/{\mathcal{F}}_P$: any finite tuple can be restricted to a neighbourhood omitting all its other support points, and its $P$-component is unaffected by such restriction. The resulting sequence on stalks isExactness of sheaves can be tested on stalks, so this proves the rational principal-parts resolution on an algebraic curveBoth right-hand sheaves are flasque: restrictions for the constant rational-section sheaf are identities between nonempty opens, and restrictions for the rational principal-parts sheaf are projections, with extension by zero. ConsequentlySurjectivity on stalks has not been confused with surjectivity on global sections.

For an explicit failure of global surjectivity take $X={\mathbb{P}}^1$ and $\mathcal{F}=\mathcal{O}(-2)$. Use coordinate $z=X_1/X_0$ and frame $e_0=X_0^{-2}$ on the finite chart. At infinity use $w=1/z$ and $e_{\infty }=X_1^{-2}=z^{-2}{e}_0$. Prescribe the rational principal part of $z^{-1}{e}_0$ at zero and zero at every other point. If a rational section $r(z)e_0$ realized this tuple, then $r-z^{-1}$ would be regular at zero and $r$ would be regular at every other finite point. Thus $r=z^{-1}+p(z)$ with $p\in k[z]$. Regularity at infinity would requireto be regular at $w=0$. Its $w^{-1}$ term cannot cancel with any term of the polynomial contribution, whose powers are at most $-2$. This is impossible. Hence the global rational-principal-part map need not be surjective.

For an affine curve it is always surjective, and here is a direct construction. Write $X=\mathrm{Spec}A$, $K=\mathrm{Frac}A$ and $\mathcal{F}=\tilde{M}$. By the affine module sheaf construction, $R=M{\otimes }_{A}K=:M_K$ and ${\mathcal{F}}_P=M_{{\mathfrak{m}}_P}$. Choose representatives $v_P\in M_K$ for a prescribed finite tuple, and choose one nonzero $a\in A$ with $a{v}_P\in M$ for all of them. The zero set $T=V(a)$ is finite. The ring $A/(a)$ is Artinian, since it is Noetherian of dimension zero, and its Artinian decomposition into local factors givesPrescribe the class of $a{v}_P$ in the indicated component for each requested point in $T$, and zero in the other components. A requested point outside $T$ already has zero principal part because $a$ is a unit there. Lift the tuple to $w\in M$. Then $s=w/a\in M_K$ satisfies $s-v_P\in M_{{\mathfrak{m}}_P}$ at every requested point in $T$, and is regular at every other point of $T$. Outside $T$ it is regular because $a$ is invertible. It therefore realizes exactly the prescribed tuple. This proves affine principal-parts interpolation for a locally free sheaf and givesNo nonsingularity assumption was used; the construction applies to singular irreducible affine curves as well.

Recall that a flasque sheaf has surjective restriction maps on every inclusion of open subsets. We first prove the flasque-kernel section-lifting lemma: in an exact sequence $0\to \mathcal{A}\to \mathcal{B}\to \mathcal{C}\to 0$ of sheaves, with $\mathcal{A}$ a flasque sheaf, every section of $\mathcal{C}$ on any open subset $V$ lifts to a section of $\mathcal{B}$ on $V$.

Fix such a section $c$. Order its lifts on open subsets $W\subseteq V$ by extension. A chain has an upper bound by the sheaf gluing axiom, so Zorn's lemma gives a maximal lift $b_W$. If $W\ne V$, choose a point outside $W$ and a neighborhood $T\subseteq V$ on which $c$ has a local lift $b_T$. On $W\cap T$, the difference $b_T-b_W$ is a section of $\mathcal{A}$. Extend it to $T$ by flasqueness and subtract it from $b_T$. The corrected lift now agrees with $b_W$ and glues on $W\cup T$, contradicting maximality. Thus $W=V$, proving the lemma.

It follows also that if both $\mathcal{A}$ and $\mathcal{B}$ are flasque sheaves, then $\mathcal{C}$ is a flasque sheaf. Given a section of $\mathcal{C}$ on $U\subseteq V$, first lift it to $\mathcal{B}$ on $U$, extend that lift to $V$, and project to $\mathcal{C}$.

Next, an injective sheaf of modules $\mathcal{I}$ is a flasque sheaf. For $U\subseteq V$, let $j_U$ and $j_V$ denote the open inclusions into $X$. Extension by zero gives a monomorphism $j_{U!}{\mathcal{O}}_U\to j_{V!}{\mathcal{O}}_V$: on every stalk it is either $0\to {\mathcal{O}}_{X,x}$, the identity, or $0\to 0$. By the adjunction for extension by zero,Injectivity therefore makes $\Gamma (V,\mathcal{I})\to \Gamma (U,\mathcal{I})$ surjective. This proves injective module sheaves are flasque.

Take an injective resolution of sheaves of $\mathcal{F}$. Write ${\mathcal{C}}^0=\mathcal{F}$ and, successively,The initial ${\mathcal{C}}^0$ is a flasque sheaf; each ${\mathcal{I}}^r$ is a flasque sheaf; and the quotient argument proves inductively that all ${\mathcal{C}}^r$ are flasque sheaves. The section-lifting lemma makes each displayed sequence exact after global sections. Consequently the complex of global sections of the injective resolution of sheaves has zero positive cohomology. Since that complex defines sheaf cohomology,

We prove the vanishing by an injective resolution of sheaves and the flasque-kernel section-lifting lemma.

For a short exact sequence of sheaves $0\to \mathcal{F}\to \mathcal{G}\to \mathcal{H}\to 0$ with flasque kernel, every section of $\mathcal{H}$ on an open set lifts to $\mathcal{G}$ there. Indeed choose a maximal partial lift by the Zorn lemma, using the sheaf gluing axiom on chains. At any point outside its domain choose a local lift. The difference on the overlap is a section of $\mathcal{F}$ and extends to the new open set by flasqueness; subtract that extension from the new lift and glue. Maximality forces the lift's domain to be the whole open set. Thus taking sections preserves this short exact sequence.

Every injective sheaf is flasque: for $U\subseteq V$, the monomorphism between the extension by zero sheaves ${\mathbb{Z}}_U\hookrightarrow {\mathbb{Z}}_V$ and the defining extension property of injectivity make the restriction $\mathcal{J}(V)\to \mathcal{J}(U)$ surjective. Embed $\mathcal{F}$ into an injective sheaf $\mathcal{J}$. The quotient $\mathcal{Q}$ is flasque as well: lift a section of $\mathcal{Q}$ on a smaller open to $\mathcal{J}$, extend it in $\mathcal{J}$, and project.

The long exact sequence in sheaf cohomology now gives $H^1(X,\mathcal{F})=0$, since $\Gamma (X,\mathcal{J})\to \Gamma (X,\mathcal{Q})$ is surjective, and for $i\ge 2$ gives $H^i(X,\mathcal{F})\cong H^{i-1}(X,\mathcal{Q})$, since injectives have zero positive-degree derived functors. Repeat the construction for the flasque quotient. After finitely many such shifts any specified positive degree reduces to a vanishing first group. HenceNo separation, compactness or scheme hypothesis is used.