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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.

Fix $p_0\in X$ on a connected compact Riemann surface, and construct its Jacobian variety by periods as in the first solution. For a divisor on an algebraic curve $E={\sum }_p{n}_{p}p$ of degree zero, defineChanging the paths adds a period lattice element; changing $p_0$ adds a common integral multiplied by $\sum n_p=0$. The Abel theorem for divisors saysEquivalently, effective divisors on an algebraic curve of the same degree have the same Abelian sum exactly when they are equivalent under linear equivalence of divisors.

We need a meromorphic version of the Riemann bilinear relations. Let $\eta$ be a differential of the third kind, with residues $n_p$ at its simple poles, and let $\omega$ be a holomorphic differential form. Choose representatives of the cycles and paths avoiding the poles in the same cut polygon, and let $F(p)={\int }_{p_0}^p\omega$ there. ThenHere the equality is an equality of the chosen complex numbers, before taking any quotient. To prove it, remove small disks about the poles. Since $d(F\eta )=\omega \wedge \eta =0$ on the remaining polygon, its boundary integral is zero. Pairing opposite outer edges gives the left side, just as in the first solution. The clockwise boundary about $p$ contributes $-2\pi i{n}_{p}F(p)$ by the residue theorem. Moving these terms to the other side proves the formula and its sign.

Such an $\eta$ exists whenever $\sum n_p=0$. Here is the required existence argument. Let $S$ be the reduced support of $E$, and considerLocally the last map takes $cdz/z$ to $c$. By Serre duality, the obstruction in $H^1(K_X)\cong \mathbb{C}$ is dual to restriction of constant functions, so it is the sum of the residues. Thus the kernel of this sum is exactly the image of global meromorphic differentials. Alternatively, the Riemann-Roch theorem gives $h^0(K_X(S))-h^0(K_X)=|S|-1$, and the residue theorem identifies that image with the same codimension-one subspace. Subtracting a linear combination of the normalized holomorphic differential forms makes $A_i(\eta )=0$ for every $i$.

Suppose now that $\mathrm{AJ}(E)=0$. Let $u_i={\sum }_p{n}_p{\int }_{p_0}^p{\omega }_i$ with the paths used above. There are integer columns $m,n$ such that $u=m+\tau n$. The normalized meromorphic bilinear formula gives $B_i(\eta )=2\pi i{u}_i$. Thereforehas $a$-periods $-2\pi i{n}_i$ and $b$-periods $2\pi i{m}_i$. Its integrals around the poles are also in $2\pi i\mathbb{Z}$. These cycles generate the homology of the punctured surface, sois a single-valued, nowhere-zero holomorphic function away from the support of $E$. Near $p$, write ${\eta }^{\prime }=n_{p}dz/z+h(z)dz$. Then $f=z^{n_p}\exp (H(z))$ times a nonzero constant, with $H^{\prime }=h$. It extends meromorphically with order exactly $n_p$. Hence $\mathrm{div}(f)=E$.

Conversely, if $E=\mathrm{div}(f)$, take $\eta =df/f$. It has residues $n_p$ and every period lies in $2\pi i\mathbb{Z}$: continuation of a local logarithm of $f$ changes it by an integral multiple of $2\pi i$. Write $A_i(\eta )=2\pi i{r}_i$ and $B_i(\eta )=2\pi i{s}_i$. Applying the meromorphic bilinear formula to ${\omega }_i$ givesThus $u\in \Lambda$ and $\mathrm{AJ}(E)=0$, completing both directions without assuming the conclusion as a property of the Jacobian variety.

For completeness, the effective degree-$d$ version lives on the symmetric product of a curve $X^{(d)}=X^d/{\mathfrak{S}}_d$. The sum of integrals on $X^d$ is invariant under the finite symmetric group and descends to the Abelian sum map $u_d$. The quotient is locally described by elementary symmetric coordinates, so repeated points are included. The proof above identifies its fibre over $u_d(D)$ withExplicitly, a nonzero global section gives its effective zero divisor, and two sections give the same divisor precisely when their quotient is constant. These maps are holomorphic in local symmetric coordinates. More intrinsically, a family of effective divisors on an algebraic curve in this fixed class determines a line of sections of ${\mathcal{O}}_X(D)$, after locally trivializing any line bundle pulled back from the parameter space. Conversely such a line of sections gives its family of zero divisors on an algebraic curve. These operations are inverse and unaffected by that trivialization, proving the identification with the projective space in families as well as on points.

Finally this description identifies the analytic Jacobian variety with ${\mathrm{Pic}}^0(X)$. Every degree-zero line bundle has a meromorphic section: after twisting by a sufficiently large effective divisor, the Riemann-Roch theorem supplies a nonzero global section. Its divisor represents the original line bundle. The correspondence to the period quotient is well-defined and injective by the theorem. It is surjective because one may choose $g$ distinct points with independent evaluations of holomorphic differential forms; the derivative of the sum of their Abel integrals is then an isomorphism. Hence the subgroup generated by point differences contains an open subset of the connected complex torus, and must be the whole group. Local integration coordinates, or the exponential sequence for ${\mathcal{O}}_X$, give the same holomorphic identification. This justifies the subsequent use of degree-$d$ line bundles as points of a translate of the Jacobian variety.

We prove the Birkhoff–Grothendieck theorem by induction on the rank of the vector bundle $E$. Rank zero is immediate; rank one is the classification $\mathrm{Pic}({\mathbb{P}}_k^1)\cong \mathbb{Z}$ established above.

First there is a largest integer $a$ with $H^0(E(-a))\ne 0$. To see existence and boundedness directly, trivialize $E$ on the two standard affine charts: the associated finite projective modules over the principal ideal domains $k[t]$ and $k[t^{-1}]$ are free. Let $A(t)\in G{L}_r(k[t,t^{-1}])$ be its transition matrix, so a section of $E(-a)$ is a pair satisfyingIf $a$ exceeds the largest exponent in any entry of $A$, the right side has only strictly negative powers; the left side has nonnegative powers. Hence both vanish. For $a$ sufficiently negative, take a nonzero constant vector $p$ and put $q=t^{a}A(t{)}^{-1}p$; all its exponents are nonpositive, so it supplies a nonzero section. Thus a largest $a$ exists.

Such a section gives a nonzero map $\mathcal{O}(a)\to E$. It is injective, since it is nonzero generically and the source is torsion-free. Saturate its image. The resulting line subbundle is $\mathcal{O}(a+d)$ for the degree $d\ge 0$ of the effective divisor of zeros of the original map. If $d>0$, its inclusion in $E$ would give a nonzero section of $E(-a-d)$, contradicting maximality. Therefore $d=0$ andhas a locally free quotient. By induction, $Q\cong {⨁}_j\mathcal{O}(b_j)$.

Twist this exact sequence by $\mathcal{O}(-a-1)$. The outer groups $H^0(\mathcal{O}(-1))$ and $H^1(\mathcal{O}(-1))$ are zero by the Čech cohomology of twists on the projective line. The long exact sequence in sheaf cohomology therefore givesThe first is zero by the choice of $a$, so each $H^0(\mathcal{O}(b_j-a-1))$ vanishes. Thus $b_j\le a$ for every $j$. The extension class lies insince $a-b_j\ge 0$. The identification of this Ext functor with cohomology can be seen without additional duality: local splittings of a vector-bundle extension differ by a Čech cocycle in $\mathcal{H}om(Q,\mathcal{O}(a))$, and a coboundary changes them to compatible splittings. Vanishing therefore gives a global splitting. Consequentlycompleting the induction over any field $k$.

The degrees can be arranged in nonincreasing order and their multiset is unique. Indeed, for any such decomposition,so the dimensions of the twisted global sections recover the number of summands of every degree.

The overlap of the two charts is obtained by inverting $c=t^3$. Since $c^2=b^3$, this also inverts $b=t^2$; then $t=c/b$ and $t^{-1}=b/c$. Thus its coordinate ring is $R[t,t^{-1}]$. The other chart has coordinate $s=t^{-1}$. A global section is therefore an element in the intersectioninside $R[t,t^{-1}]$: a Laurent polynomial with both only nonnegative and only nonpositive exponents is constant.

Let $k=R/(\pi )$. Reducing the presentation of $B$ givesAs a vector space and as a ring extension, this iswhere $a^2=0$ and $a$ is annihilated by $b,c$. In particular $a\ne 0$: the free basis of $B$ in part ii remains a basis after tensoring with $k$. Although $a$ mapped to $\pi t$ before reduction, reducing the subring is not the same operation as taking its image inside $k[t]$; that distinction is exactly where the extra section arises.

On the overlap, $c$ is invertible and $ac=0$, so $a$ restricts to zero. Therefore the section $a$ on the first chart glues to zero on the second chart and defines a nonzero global nilpotent section $\eta$ on $X_s$. More precisely the reduced first-chart ring is $k[t^2,t^3]$, andin $k[t,t^{-1}]$. The only extra compatible sections are multiples of $a$. HenceThe map printed in the question reduces $R$ modulo $\pi$ and includes its image as constants. Its kernel is $\pi R$ and its cokernel is $k\eta$. More strongly, even the usual base-change map after tensoring,is not surjective. This is global-section base-change failure in a flat projective family, despite the properness and flatness proved above.

A locally ringed space is a topological space $X$ equipped with a sheaf of commutative rings ${\mathcal{O}}_X$ whose stalk ${\mathcal{O}}_{X,x}$ at every point is a local ring. A morphism of locally ringed spaces $(f,f^{\#}):(Y,{\mathcal{O}}_Y)\to (X,{\mathcal{O}}_X)$ consists of a continuous map $f:Y\to X$ and a morphism of sheaves of rings $f^{\#}:{\mathcal{O}}_X\to f_{\ast }{\mathcal{O}}_Y$, with every induced stalk map ${\mathcal{O}}_{X,f(y)}\to {\mathcal{O}}_{Y,y}$ a local homomorphism: the inverse image of the target maximal ideal is the source maximal ideal. Here $f_{\ast }{\mathcal{O}}_Y$ is the direct image sheaf, with sections ${\mathcal{O}}_Y(f^{-1}U)$ on $U$.

Let $\varphi :A\to B$ be a ring homomorphism. Define the map between the spectra of rings byThe inverse image is a prime ideal, and $f^{-1}(D(a))=D(\varphi (a))$. Thus $f$ is a continuous map, because the principal open subsets form a basis. On these principal open subsets, define the structure sheaf map using the localizationsThese maps commute with restrictions and consequently give a morphism of sheaves. At $\mathfrak{q}$, putting $\mathfrak{p}={\varphi }^{-1}\mathfrak{q}$, the stalk map is $A_{\mathfrak{p}}\to B_{\mathfrak{q}}$. An element $b/s$ belongs to its inverse image of $\mathfrak{q}{B}_{\mathfrak{q}}$ exactly when $b\in \mathfrak{p}$, so this is a local homomorphism. We have constructed a morphism of locally ringed spaces.

Conversely, take a morphism of locally ringed spaces $f:\mathrm{Spec}B\to \mathrm{Spec}A$. Its map on global sections gives a ring homomorphismLocality of the stalk map implies, for every $a\in A$,Hence the point map must be $\mathfrak{q}\mapsto {\varphi }^{-1}\mathfrak{q}$. On $D(a)$, its structure sheaf map must send $b/1$ to $\varphi (b)/1$ and the inverse of $a$ to the inverse of $\varphi (a)$. Thus it is precisely the localization map constructed above. Agreement on this basis proves agreement of the morphisms of sheaves. In particular, the inducing ring homomorphism exists and is unique.

If $\varphi$ is surjective, write $B=A/I$. The correspondence between prime ideals of $A/I$ and prime ideals of $A$ containing $I$ identifies $\mathrm{Spec}B$ homeomorphically with the closed subset $V(I)$: the principal open subsets correspond on both sides. Moreover, the stalk maps are the surjections $A_{\mathfrak{p}}\to (A/I{)}_{\mathfrak{p}/I}$, so ${\mathcal{O}}_X\to f_{\ast }{\mathcal{O}}_Y$ is surjective.

For the converse, it is important not to assume that global sections preserve an arbitrary surjection of sheaves. Here the affine module-sheaf equivalence supplies the extra information: on every $D(a)$,Thus $f_{\ast }{\mathcal{O}}_Y$ is the quasi-coherent sheaf associated with the $A$-module $B$, and the cokernel sheaf of ${\mathcal{O}}_X\to f_{\ast }{\mathcal{O}}_Y$ is associated with the module $C=\mathrm{coker}(A\to B)$, since localization of a module is exact. Surjectivity of the morphism of sheaves gives $C_{\mathfrak{p}}=0$ for every prime ideal $\mathfrak{p}$. This implies $C=0$: if $0\ne c\in C$, its proper annihilator lies in a maximal ideal $\mathfrak{m}$, and then $c/1\ne 0$ in $C_{\mathfrak{m}}$. Therefore $A\to B$ is surjective. We have proved the closed immersion criterion for affine schemes:

Use the local-presentation definition of a quasi-coherent sheaf: locally it admits a presentation ${\mathcal{O}}^{(I)}\to {\mathcal{O}}^{(J)}\to \mathcal{F}\to 0$ by direct sums of copies of the structure sheaf, with no finiteness requirement on the index sets. If $M$ is an $A$-module, choose a free module presentation $A^{(I)}\to A^{(J)}\to M\to 0$. Exactness of localization of a module gives the corresponding local presentation of $\tilde{M}$. Hence a sheaf which is associated with a module on every affine open subset is a quasi-coherent sheaf.

For the converse, begin with a quasi-coherent sheaf and a local presentation on an affine open subset $V=\mathrm{Spec}R$. The free sheaves have global sections $R^{(I)}$ and $R^{(J)}$: a section has locally finite support, and quasi-compactness of $V$ makes its support finite globally. Part (i) identifies the presentation map with an $R$-module homomorphism. Exactness of localization then identifies its cokernel sheaf with ${\tilde{M}}_V$ for the module cokernel $M_V$. Thus there is an affine open cover on which $\mathcal{F}$ is associated with modules.

It remains to show this on an arbitrary affine open subset $U=\mathrm{Spec}A$, rather than only the selected cover. Around each point of $U$, choose an affine neighborhood contained in $U$ with such a local presentation. Choose a finite cover $U={\bigcup }_{i=1}^{r}D(s_i)$ refining these neighborhoods. On each $D(s_i)$ the restriction is associated with a module. Its restriction to $D(s_i{s}_j)$ is therefore the corresponding localization, and the same is true after intersecting with $D(a)$. Put $M=\Gamma (U,\mathcal{F})$. The sheaf gluing axiom giveswhere the arrow is the difference of the two restriction maps. Localization of a module commutes with kernels and finite products. Localizing this formula at $a$ therefore gives exactly the gluing formula for $D(a)$, covered by the $D(a{s}_i)$. Consequentlycompatibly with restriction. These isomorphisms on the principal open subsets identify $\tilde{M}$ with $\mathcal{F}{|}_U$. We have proved the global-section localization for quasi-coherent sheaves and, in particular,

An $A$-module homomorphism $h:M\to N$ induces, by localization of a module, compatible mapson all principal open subsets $D(a)$. These glue to an ${\mathcal{O}}_X$-linear morphism of sheaves $\tilde{h}:\tilde{M}\to \tilde{N}$.

Conversely, taking global sections of an ${\mathcal{O}}_X$-linear morphism of sheaves gives an $A$-module homomorphism $h:M\to N$, since $\Gamma (X,\tilde{M})=M$ and $\Gamma (X,\tilde{N})=N$. On $D(a)$, compatibility with restriction determines the image of $m/1$, and ${\mathcal{O}}_X$-linearity determines the image of $a^{-r}m$. It must be $h(m)/a^r$. Thus the morphism of sheaves is uniquely recovered from $h$. Both constructions preserve addition and composition, proving the affine module-sheaf equivalence on morphisms:No finite-generation assumption is used.

Take the projective line $X={\mathbb{P}}_k^1$ and the coherent sheaves $\mathcal{F}={\mathcal{O}}_X$ and $\mathcal{G}={\mathcal{O}}_X(-1)$, where the latter is a twisting sheaf on projective space. Both are line bundles, so each stalk is a free rank-one ${\mathcal{O}}_{X,x}$-module and therefore ${\mathcal{F}}_x\simeq {\mathcal{G}}_x$ at every point.

They are nevertheless not isomorphic. On the standard charts with coordinates $t$ and $t^{-1}$, choose frames $e_0,e_1$ of $\mathcal{O}(-1)$ with $e_1=t^{-1}{e}_0$. A global section would give polynomials $a(t)$ and $b(t^{-1})$ satisfyingThe left side has only nonnegative powers of $t$, and the right side only strictly negative powers. Thus both vanish, giving $\Gamma (X,\mathcal{O}(-1))=0$. For the structure sheaf, the corresponding equality is $a(t)=b(t^{-1})$, whose solutions are precisely the constants. HenceA morphism of sheaves which is an isomorphism on all stalks would be an isomorphism; this example only asserts the existence of separate stalk isomorphisms, not that they come from one morphism of sheaves.

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,

Take the affine scheme $X={\mathbb{A}}_k^1$, its function field $K=k(t)$, and the constant sheaf $\mathcal{K}$ with value $K$. By Question 4(ii), $\mathcal{K}$ is a flasque sheaf. At the two distinct closed rational points $p_0=0$ and $p_1=1$, let $S_0,S_1$ be skyscraper sheaves with value $K$: on an open subset $V$, $S_i(V)=K$ if $p_i\in V$ and is zero otherwise. Both are flasque sheaves.

There is a natural morphism of sheaveswhich sends a constant rational function to the same element in every summand present on the open subset. At $p_i$, its stalk map is the identity $K\to K$, and at any other point the target stalk is zero. Thus $\rho$ is a surjection of sheaves, although it is not surjective on global sections. Put $\mathcal{F}=\ker \rho$.

This can even be an example of sheaves of ${\mathcal{O}}_X$-modules: let regular functions act on each copy of $K$ by multiplication after embedding them in $K$. At the closed points this action is multiplication by a rational function, not evaluation at the point. With these actions, $\rho$ is ${\mathcal{O}}_X$-linear.

The exact sequence $0\to \mathcal{F}\to \mathcal{K}\to S_0\oplus S_1\to 0$, together with flasque acyclicity, givesHence $H^0(X,\mathcal{F})=0$ and $H^1(X,\mathcal{F})=K^2/\Delta K\simeq K\ne 0$. In contrast, use the one-member affine open cover $\mathcal{U}=\{X\}$. Its alternating Čech cochain complex has no positive-degree terms, so ${\check{H}}^1(\mathcal{U},\mathcal{F})=0$. We have the explicit discrepancyThe sheaf $\mathcal{F}$ is not a quasi-coherent sheaf, as also follows from its nonzero higher sheaf cohomology on an affine scheme. Thus this example does not contradict the acyclic cover theorem for quasi-coherent sheaves on affine covers.

Let $f:{\mathbb{P}}_k^1\to \mathrm{Spec}k$ be the structure map, a projective morphism. Let $D$ be the union of the two rational points $[0:1]$ and $[1:0]$. The homogeneous polynomial $t_0{t}_1$ defines this effective Cartier divisor and gives the exact sequence of coherent sheavesThe twisting sheaf on projective space $\mathcal{O}(-2)$ has no nonzero global sections: the chart calculation of Question 3(iv) now requires $a(t)=t^{-2}b(t^{-1})$, again impossible for nonzero polynomials. We also have $\Gamma ({\mathbb{P}}^1,\mathcal{O})=k$ and $\Gamma (D,{\mathcal{O}}_D)=k\oplus k$.

The direct image sheaf under $f$ is the module of global sections on the one-point scheme. Consequently the induced sequence isIts last nonzero map has cokernel isomorphic to $k$, since $(a,b)\mapsto b-a$ has the diagonal as its kernel. Thus the induced sequence is not exact at $f_{\ast }{\mathcal{O}}_D$, proving that projective direct image need not preserve surjections.

A ring homomorphism is understood to preserve $1$. Start with $\phi :A\to B$. Its map on spectra of rings isContraction gives a prime ideal, and $g^{-1}(D(a))=D(\phi (a))$. Thus $g$ is continuous for the Zariski topology. On each principal open subscheme, the structure sheaf map is the ring homomorphismThese maps commute with restrictions, so they define a sheaf morphism ${\mathcal{O}}_X\to g_{\ast }{\mathcal{O}}_Y$. At $\mathfrak{q}$, with $\mathfrak{p}={\phi }^{-1}(\mathfrak{q})$, its stalk map is $A_{\mathfrak{p}}\to B_{\mathfrak{q}}$. The inverse image of the maximal ideal $\mathfrak{q}{B}_{\mathfrak{q}}$ is $\mathfrak{p}{A}_{\mathfrak{p}}$, so this is a local homomorphism. We have constructed a morphism of locally ringed spaces.

Conversely, let $(g,g^{\#})$ be a morphism of locally ringed spaces. Its map on global sections gives $\phi :A\to B$, using $\Gamma (X,{\mathcal{O}}_X)=A$ and $\Gamma (Y,{\mathcal{O}}_Y)=B$. Fix $\mathfrak{q}\in Y$ and put $\mathfrak{p}=g(\mathfrak{q})$. Compatibility with the stalk maps and the local homomorphism property giveConsequently $\mathfrak{p}={\phi }^{-1}(\mathfrak{q})$, so the underlying map is forced. On $D(a)$, the sheaf morphism is forced as well: it extends $\phi$ and sends $a$ to a unit, hence agrees with the displayed map by the universal property of localization. The principal open subschemes form a basis, so the entire sheaf morphism is determined. Taking global sections of the construction recovers $\phi$. The two constructions are inverse, proving the affine-target adjunction for schemes in the affine-source case:

To describe the real affine plane scheme points, write $R=\mathbb{R}[t_1,t_2]$. It is a unique factorization domain of Krull dimension two. Its points are exactly the following prime ideals:

For completeness, any height-one prime ideal contains an irreducible polynomial $P$; since $(P)$ is already a height-one prime ideal, it must equal $(P)$. Every remaining nonzero prime ideal has height two and is maximal, by Krull dimension. For a nonreal pair, evaluation generates all of $\mathbb{C}$ over $\mathbb{R}$, so its kernel is maximal. Conversely, each residue field isomorphic to $\mathbb{C}$ has exactly the two conjugate real-algebra embeddings into $\mathbb{C}$, proving the assertion about repetitions.

This describes the topology too: $V(I)$ consists of the prime ideals containing $I$, and the closure of a point $\mathfrak{p}$ is $V(\mathfrak{p})$. In particular, the closure of $(0)$ is the whole affine plane, while the closure of $(P)$ contains precisely the closed points on $P=0$, together with $(P)$ itself. The spectrum is much larger than the set ${\mathbb{R}}^2$. For example, $(t_1^2+1)$ is a height-one point although its curve has no real points. Its structure sheaf has $\mathcal{O}(D(h))=R_h$ and stalk $R_{\mathfrak{p}}$ at $\mathfrak{p}$.

The induced morphism of schemesis contraction of prime ideals, with the structure sheaf maps given above. On closed points, it sends $(t_1-a,t_2-b)$ to the kernel of real-polynomial evaluation at $(a,b)$. A real closed point has one complex point above it; a nonreal closed point has two, interchanged by complex conjugation. The source generic point maps to the target generic point. The source height-one points are generated by irreducible complex polynomials $Q$. Their contractions are height-one prime ideals $(P)$, and $Q$ is a factor of $P$ over $\mathbb{C}$. An irreducible real $P$ either stays irreducible over $\mathbb{C}$ or splits into two distinct conjugate irreducible factors. Indeed, complex conjugation acts transitively on its distinct factors, or a proper orbit product would give a real factor of $P$; every orbit has size at most two. Repeated factors are excluded by separability in characteristic zero. Thus one or two height-one points lie above $(P)$.

The complexification fibres of a real scheme give a uniform description of all scheme-theoretic fibres, including the nonclosed points, is especially useful. The extension of coordinate rings isa free $R$-module with basis $1,s$. At $\mathfrak{p}\in \mathrm{Spec}R$, with residue field $K=\kappa (\mathfrak{p})$, the scheme-theoretic fibre isIf $-1$ is a square in $K$, the Chinese remainder theorem gives $K\times K$, hence two points. Otherwise it is a quadratic field extension, hence one point. The polynomial has no repeated root in characteristic zero, so all these scheme-theoretic fibres are reduced schemes. This also proves surjectivity. Conjugation acts on each two-point fibre by exchanging its points and fixes each one-point fibre. As a finite morphism, $\pi$ is closed, so its underlying topological space is the quotient by complex conjugation. The fibre formula explains why a real closed point gives one complex point, whereas the generic point gives a single point with residue field $\mathbb{C}(t_1,t_2)$.

The point to retain in the direct-image theorem is quasi-compactness of inverse images of affine opens and of overlaps; no separation assumption has been supplied. The underlying space of a Noetherian scheme is Noetherian, so every open subset of $X$ is quasi-compact. Fix an affine open subscheme $V=\mathrm{Spec}A$ of $Y$, put $W=f^{-1}V$, and choose a finite open cover by affine open subschemes $W={\bigcup }_i{U}_i$. For each pair $i,j$, choose a finite affine cover $(W_{ij\ell }{)}_{\ell }$ of $U_i\cap U_j$. Empty overlaps contribute no terms.

Write $M=\Gamma (W,\mathcal{F})$. The sheaf gluing axiom gives the exact sequencewhere $\delta$ is the difference of the two restrictions to each overlap chart. All terms are $A$-modules through $f$. For $a\in A$, the inverse image of $D(a)$ cuts each $U_i$ and $W_{ij\ell }$ by a principal open subscheme. Because $\mathcal{F}$ is a quasi-coherent sheaf, sections on these smaller affine charts are the corresponding module localizations at $a$. Exactness of localization and its commutation with finite products now identify the localized equalizer with the equalizer for the restricted cover. ThusThe isomorphisms respect restrictions, proving $(f_{\ast }\mathcal{F}){|}_V\cong \tilde{M}$ on the basis of principal open subschemes. Since $V$ was arbitrary, $f_{\ast }\mathcal{F}$ is a quasi-coherent sheaf. The finite covers of overlaps are what allow the proof to work for nonseparated Noetherian schemes; this is the quasi-coherence of direct image under a quasi-compact quasi-separated morphism.

For a quasi-coherent sheaf which is not coherent but has coherent direct image, use $f:{\mathbb{P}}_k^1\to \mathrm{Spec}k$ andThis infinite negative-twist sum with zero global sections is quasi-coherent: on each standard affine chart it is the sheaf associated with a direct sum of free rank-one modules. Its stalk at every point, modulo the maximal ideal, is an infinite-dimensional vector space. A finitely generated module would have a finite-dimensional quotient, so $\mathcal{F}$ is not a coherent sheaf.

Nevertheless, $\Gamma ({\mathbb{P}}^1,\mathcal{O}(-1))=0$. This follows from cohomology of twisting sheaves on projective space, or directly by gluing on the two standard affine charts: if $z=t_1/t_0$, a section is a polynomial $p(z)$ on the first chart and $q(z^{-1})$ on the second with $p(z)=z^{-1}q(z^{-1})$, which forces both to vanish. Global sections commute with this direct sum: they are the kernel of the difference map for the two-chart cover, and direct sums commute with that finite equalizer of modules. Hencewhich is coherent on $\mathrm{Spec}k$. Both schemes in this example are Noetherian.

For a finite morphism, such an example is impossible. On an affine open subscheme $V=\mathrm{Spec}A$ of the target, its inverse image is $\mathrm{Spec}B$, with $B$ a finite $A$-module. Write $\mathcal{F}{|}_{f^{-1}V}=\tilde{M}$. Its direct image corresponds to $M$ considered as an $A$-module. If the direct image is coherent, $M$ is finitely generated over $A$. The same generators also generate it over $B$, because $A$ acts through $B$. Since $B$ is Noetherian, $\tilde{M}$ is coherent. Conversely, a finite set of $B$-generators combined with a finite set of $A$-generators of $B$ gives finitely many $A$-generators of $M$. Thus coherence reflected by finite direct image gives the stronger equivalence

Finally, let $Y=\mathrm{Spec}k[x,y]$, let $X=Y\setminus \{(x,y)\}$ be the punctured affine plane, and let $j:X\hookrightarrow Y$ be the open immersion. Both are integral schemes, but $j$ is not an isomorphism of schemes because it omits a point. The cover $X=D(x)\cup D(y)$ givesinside the field of fractions $k(x,y)$. Indeed, in a reduced fraction, membership in the first localization forces every denominator factor to be associated to $x$, while membership in the second forces it to be associated to $y$. Unique factorization and coprimality force the denominator to be a unit. By the theorem just proved, $j_{\ast }{\mathcal{O}}_X$ is quasi-coherent on the affine $Y$, hence determined by this module of global sections. The natural map ${\mathcal{O}}_Y\to j_{\ast }{\mathcal{O}}_X$ corresponds to the identity of $k[x,y]$, soa coherent sheaf. This is a concrete case of codimension-two extension of regular functions on a normal variety.

The intersection hypothesis says that $X$ is a semi-separated scheme. For an affine open subscheme $V\subseteq X$, the intersection $V\cap U$ is affine. Restricting the given short exact sequence of sheaves to it and applying the affine module-sheaf equivalence gives an exact sequence of global sections:These are exactly the sections on $V$ of the three direct image sheaves. Affine opens form a basis, and every section of the last sheaf on such a basis open lifts to the middle sheaf. This proves surjectivity as a sheaf morphism; left exactness of direct image supplies the other positions. Hence direct image preserves this short exact sequence. The crucial ingredient is exactness of sections of quasi-coherent sheaves on an affine intersection; arbitrary open immersions need not have this property.

For the cohomology comparison, every nonempty finite intersectionis affine. This follows by mathematical induction, intersecting the affine intersection already obtained with the next affine open. The restriction of $\mathcal{F}$ to it is quasi-coherent, so vanishing of quasi-coherent cohomology on an affine scheme givesWe now prove why this local vanishing gives the acyclic cover theorem, rather than identifying the two sorts of cohomology without a comparison.

Take a flasque resolution $0\to \mathcal{F}\to {\mathcal{I}}^0\to {\mathcal{I}}^1\to \cdots$. Here a flabby sheaf has surjective restriction maps; its restrictions to open subsets are still flabby and have zero higher sheaf cohomology. Form the double complexThe horizontal differential is the alternating restriction map $d_{\mathrm{C}}$; the vertical one $d_{\mathrm{I}}$ is induced by the resolution. They commute, so the total differential in bidegree $(p,q)$ is $d_{\mathrm{C}}+(-1{)}^p{d}_{\mathrm{I}}$ and has square zero. The Čech resolution on a semi-separated scheme uses precisely these intersections.

A flabby sheaf has zero positive Čech cohomology for a finite open cover, and its degree-zero Čech cohomology is its global sections. One way to establish this auxiliary fact is to use the exact augmented two-open complexExactness at the first two terms is the sheaf gluing axiom; the last map is onto because a section on the intersection extends to $V$. For the induction step, write $V$ for the union of all but the last open and $W$ for the last open. Separate Čech cochains according to whether their index list contains the last index. The resulting two-block complex compares the smaller cover of $V$ with its restricted cover of $V\cap W$, together with $\Gamma (W,\mathcal{I})$ in degree zero. By induction those two smaller cover complexes have cohomology only in degree zero, where they give $\Gamma (V,\mathcal{I})$ and $\Gamma (V\cap W,\mathcal{I})$. The displayed two-open exact sequence then gives zero positive-degree cohomology for the full cover. This proves the auxiliary fact by induction on the number of opens. Thus horizontal cohomology of $C^{\bullet ,q}$ consists only of $\Gamma (X,{\mathcal{I}}^q)$ in degree zero. Computing the cohomology of the total complex first horizontally therefore gives $H^n(X,\mathcal{F})$, by the resolution principle for sheaf cohomology.

On the other hand, vertical cohomology issince the restricted flasque resolutions compute cohomology on each intersection. The already established affine vanishing makes all rows with $q>0$ zero. The surviving row $q=0$ is exactly the Čech cochain complex ${\check{C}}^{\bullet }(\mathcal{U},\mathcal{F})$. Computing total cohomology first vertically therefore gives ${\check{H}}^n(\mathcal{U},\mathcal{F})$. These two computations are justified by the two filtrations of the first-quadrant double complex: in every total degree only finitely many terms occur, and the cover also bounds the horizontal degree. Their edge maps give the natural identificationFor negative degrees both groups are zero by convention. In particular, degree zero is the usual identification by the sheaf gluing axiom, not merely a comparison of dimensions.