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The direct sum of sheaves of abelian groups is the sheafification of the presheaf $U\mapsto {⨁}_i{\mathcal{F}}_i(U)$. It need not equal that presheaf: sections are locally represented using finitely many summands, while globally infinitely many can occur. For point-supported skyscraper sheaves at closed points, allowable point supports are a locally finite family of subsets.

For a closed point $P$ with residue field $k(P)$, evaluation gives a morphism from the structure sheaf onto the skyscraper sheaf $k_P$. Its kernel is the ideal sheaf of $P$. On a Noetherian scheme it is a coherent ideal sheaf.

This sheaf consists of holomorphic functions vanishing at $p$ on opens containing $p$, and all holomorphic functions on opens not containing $p$. Evaluation gives $0\to {\mathcal{I}}_p\to {\mathcal{O}}_M\to {\mathbb{C}}_p\to 0$, with a skyscraper sheaf as quotient. On an elliptic curve, $H^0({\mathcal{I}}_p)=0$, $H^1({\mathcal{I}}_p)\cong \mathbb{C}$, and all higher cohomology vanishes.

The printed vanishing claim needs the restriction $i\ge 1$. On $C={\mathbb{P}}^1$ with $\mathcal{F}=\mathcal{O}$ and $D=n\{\infty \}$, Question 3 gives $h^0(\mathcal{F}(D))=n+1$ for every $n\ge 0$. Thus no bound on the degree of a divisor can force vanishing in degree zero. We prove finiteness in every degree and the corrected vanishing in every positive degree, uniformly over all divisors on an algebraic curve of large degree.

First take a finite morphism $\pi :C\to {\mathbb{P}}^1$ as in Question 3 and put $E={\pi }_{\ast }\mathcal{F}$. The direct image sheaf $E$ is coherent: on an affine chart its module is finitely generated over a finite algebra and hence over the chart's coordinate ring. The two affine-cover Čech cochain complexes for $\mathcal{F}$ on $C$ and for $E$ on ${\mathbb{P}}^1$ are identical. Therefore

We need an elementary presentation of any coherent sheaf $E$ on the projective line. Choose finitely many generators on each of its two standard affine charts. A generator on the first chart restricts to an element of the second chart's module localized at $t^{-1}$. Multiplication by a sufficiently large power of $t^{-1}$ brings that element into the second chart's module. Since the transition of $\mathcal{O}(n)$ is $t^n$, this makes the original generator extend to a global section of $E(n)$. A generator on the second chart similarly extends after clearing its denominator at $t=0$. Use one common exponent for all the finitely many generators. The extended sections generate $E(n)$ on both charts. Untwisting gives a surjectionwith coherent kernel $K$. The long exact sequence in sheaf cohomology and Question 3 show that $H^1(E)$ is a quotient of the finite-dimensional $H^1(P)$, since $H^2(K)=0$. Apply the same presentation argument to $K$ to obtain finiteness of $H^1(K)$. Then the segment $H^0(P)\to H^0(E)\to H^1(K)$ proves finiteness of $H^0(E)$. Higher groups vanish by Question 3. Hence every $H^i(C,\mathcal{F})$ is finite-dimensional.

Next establish a useful degree estimate without using the duality requested later. A global regular function on $C$ is constant: a nonconstant one would give a finite morphism $C\to {\mathbb{P}}^1$ by Question 1. Such a morphism is surjective, because its closed image contains the generic point, but a regular function misses $\infty$, a contradiction. Thus $H^0(C,{\mathcal{O}}_C)=k$. Write $g=h^1(C,{\mathcal{O}}_C)$, the genus of a smooth projective curve. For any divisor on an algebraic curve $A$ and point $P$, the local discrete valuation ring givesThe last term is a length-one skyscraper sheaf. Additivity of the Euler characteristic of a coherent sheaf and repeated addition or subtraction of points giveThe inequality follows because $h^1$ is nonnegative, and is sufficient for the following argument.

Fix $P\in C$. For $n\ge g+1$ this estimate gives at least two independent sections of ${\mathcal{O}}_C(nP)$, one of them the constant section. A nonconstant section is a rational function on an algebraic variety whose poles are confined to $P$. It must have a pole there, since a globally regular function is constant. Its extension $C\to {\mathbb{P}}^1$ is finite, and its inverse image of ${\mathbb{A}}^1$ is exactly $C\setminus \{P\}$. Consequently $U=C\setminus \{P\}$ is affine.

Let $T$ be the torsion subsheaf of $\mathcal{F}$ and let $V=\mathcal{F}/T$. The support of $T$ is finite; such a coherent sheaf is a finite direct sum of sheaves supported at points and has zero higher sheaf cohomology. The torsion-free sheaf $V$ is locally free: a finitely generated torsion-free module over a discrete valuation ring is free. Tensoring by any invertible sheaf is exact, so

The inclusions $V(nP)\subset V((n+1)P)$ have point-supported quotients. Their long exact sequences in sheaf cohomology therefore give surjections on $H^1$. Moreovera section regular away from $P$ has some finite pole order at $P$. Compute this direct limit on a finite affine cover of $C$. All its intersections with $U$ are affine by Question 2, and localization commutes with filtered direct limits. The Čech cochain complex consequently givesThe maps from $H^1(C,V)$ to these groups are surjective. Choose a finite basis of this starting vector space. Each basis vector becomes zero at some stage in the direct limit; one common stage kills the entire basis. Thus there exists $n_0\ge 0$ with $H^1(C,V(n_{0}P))=0$.

Now let $D$ be any divisor on an algebraic curve with $\mathrm{deg}D\ge n_0+g$. The estimate above gives $h^0({\mathcal{O}}_C(D-n_{0}P))\ge 1$. A nonzero section identifies $D-n_{0}P$ with an effective divisor $E$ up to linear equivalence of divisors. Hence $V(D)\cong V(n_{0}P+E)$. Adding $E$ gives a point-supported quotient of $V(n_{0}P+E)/V(n_{0}P)$, so its first sheaf cohomology is a quotient of $H^1(C,V(n_{0}P))=0$. We have proved uniform high-degree vanishing on a smooth projective curve:This bound concerns all high-degree divisors, not only multiples of a previously chosen divisor.

Use the finite map $f:C\to {\mathbb{P}}^1$ from the preceding solution. The inverse images of the two standard affine charts and of their intersection are affine, because $f$ is finite. The vanishing of quasi-coherent cohomology on an affine scheme therefore lets their two-term Čech cochain complex compute $H^i(C,E)$. It has terms only in degrees zero and one, provingThis actually applies to every quasi-coherent sheaf on $C$.

For a smooth projective curve with $H^0(C,{\mathcal{O}}_C)=k$, define its genus of a smooth projective curve byThe preceding Serre duality also gives $g={\dim }_k{H}^0(C,{\Omega }_C^1)$. The condition on constants holds in particular for a geometrically connected smooth projective curve. It makes $g$ equal to the arithmetic genus $1-\chi ({\mathcal{O}}_C)$. We state the theorem with this usual convention and then give the formula valid without that condition.

For a divisor on an algebraic curve $D$ and a canonical divisor $K$ with $\mathcal{O}(K)\cong {\Omega }_C^1$, the Riemann-Roch theorem isThe degree over an arbitrary field is weighted by residue degrees: ${\mathrm{deg}}_{k}D={\sum }_p{n}_p[k(p):k]$ when $D={\sum }_p{n}_{p}p$. Equivalently, for a line bundle $L$,We now prove it via Riemann-Roch via elementary modifications.

For any closed point $p$ and any line bundle $L$, the exact sequencehas a skyscraper sheaf as its quotient, with $k$-dimension $[k(p):k]$. This follows locally from a uniformizer of the discrete valuation ring at $p$: a local line-bundle generator modulo that uniformizer spans one copy of $k(p)$. The quotient has zero higher cohomology, as it is the direct image from a finite affine scheme. The long exact sequence in sheaf cohomology and the proved higher vanishing giveEvery line bundle has a nonzero rational section of a line bundle, whose local orders give a divisor on an algebraic curve $D$ with $L\cong \mathcal{O}(D)$. Iterating the displayed identity, adding or subtracting closed points with their multiplicities, givesFor the constants convention above, $\chi ({\mathcal{O}}_C)=1-g$. The dualizing sheaf on a smooth projective curve gives $h^1(\mathcal{O}(D))=h^0(\mathcal{O}(K-D))$, which converts this Euler-characteristic identity into the stated Riemann-Roch theorem. This proves the formula over arbitrary $k$, without assuming the closed points are $k$-rational.

For vector bundles, the corresponding formula isTo see it, saturate a rational rank-one subspace to a line subbundle and induct on rank. Both Euler characteristic of a coherent sheaf and the degree of the determinant are additive in the resulting short exact sequence. The line-bundle formula just proved starts the induction.

Taking $L={\Omega }_C^1$ and using duality gives $h^0({\Omega }_C^1)=g$, $h^1({\Omega }_C^1)=1$, and hence ${\mathrm{deg}}_k{\Omega }_C^1=2g-2$. In particular a line bundle of degree greater than $2g-2$ has zero first cohomology: its dual twisted by ${\Omega }_C^1$ has negative degree, and a negative-degree line bundle has no nonzero section, since any section would have an effective divisor of zeros.

If the term “curve over $k$” is used without requiring $H^0({\mathcal{O}}_C)=k$, the unconditional formula we proved isWriting $g=h^1({\mathcal{O}}_C)$ then replaces $1-g$ by $h^0({\mathcal{O}}_C)-g$. For example, ${\mathbb{P}}_{k^{\prime }}^1$ viewed as a $k$-curve for a finite separable extension $k^{\prime }/k$ has $h^0({\mathcal{O}}_C)=[k^{\prime }:k]$, so a literal constant term $1$ would be incorrect. Alternatively the arithmetic genus $p_a=1-\chi ({\mathcal{O}}_C)$ gives the formula with constant $1-p_a$. The constants hypothesis for the usual genus is recorded in Stacks Project, genus of a curve.

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.

For a smooth projective curve $C$, let $g=h^1(C,{\mathcal{O}}_C)$ be its geometric genus, let $K_C$ be a canonical divisor, and write $\ell (D)=h^0(C,{\mathcal{O}}_C(D))$ for the dimension of a vector space of the Riemann-Roch space. The Riemann-Roch theorem saysfor every divisor on an algebraic curve $D$ on $C$.

Here is the cohomological proof of Riemann-Roch for curves. At a closed point $P$, the local ring is a discrete valuation ring. A local uniformizer shows that increasing the allowed pole order by one gives the short exact sequence of sheavesThe last term is a length-one skyscraper sheaf; its precise identification with $k$ is noncanonical, but its sheaf cohomology has $h^0=1$ and $h^1=0$. All the relevant groups of sheaf cohomology are finite-dimensional because $C$ is a projective variety, and coherent sheaf cohomology above degree one vanishes on a algebraic curve. The long exact sequence in sheaf cohomology therefore givesSince $k$ is an algebraically closed field, every closed point has degree one. Iterating this identity for both positive and negative coefficients of $D$ provesEvery global regular function on a projective variety which is an irreducible variety is constant; hence $h^0(C,{\mathcal{O}}_C)=1$ and $\chi ({\mathcal{O}}_C)=1-g$.

Finally apply Serre duality for a smooth projective curve:Taking dimensions of a vector space and inserting this in the Euler characteristic of a coherent sheaf identity proves the displayed Riemann-Roch theorem. The substantial inputs from sheaf cohomology are finiteness and vanishing for coherent sheaves on a projective curve, and Serre duality; the change of Euler characteristic of a coherent sheaf is proved directly by the point exact sequence. In particular the argument handles negative as well as effective divisors on an algebraic curve.

Let ${\mathcal{I}}_Y$ be the coherent ideal sheaf of $Y$, and let ${\mathfrak{m}}_P$ be the ideal sheaf of a closed point. Because $P\notin Y$, the stalk $({\mathcal{I}}_Y{)}_P$ is ${\mathcal{O}}_{X,P}$. Evaluation at $P$ therefore gives a surjective morphism of sheaves ${\mathcal{I}}_Y\to k_P$ to the skyscraper sheaf at $P$. Its kernel $\mathcal{J}={\mathcal{I}}_Y\cap {\mathfrak{m}}_P$ is again a coherent ideal sheaf. Fromand $H^1(X,\mathcal{J})=0$, the long exact sequence in sheaf cohomology shows that $\Gamma (X,{\mathcal{I}}_Y)\to k$ is onto. Choose $f$ mapping to $1$. It vanishes on $Y$ and satisfies $f(P)=1$. Hence $X_f\subseteq U$, and inside the affine variety $U$ it is the principal open subset defined by $f{|}_U$. A principal open of an affine variety is affine. This gives affine principal neighbourhoods from ideal-sheaf vanishing.

For a point $x\in X$ and an abelian group $A$, the skyscraper sheaf $i_{\ast }A$, with $i:\{x\}\hookrightarrow X$, isRestrictions are identities or the map to zero, so it is a flasque sheaf. The global sections equal $A$ and higher cohomology vanishes:No closed-point assumption is needed for this calculation of skyscraper sheaf cohomology. On an arbitrary topological space, its nonzero stalks are at points of $\overline{\{x\}}$ when $A\ne 0$; only for a closed point is the nonzero stalk confined to $x$.

Let $\mathcal{P}={\mathcal{M}}_X/{\mathcal{O}}_X$, the sheaf of meromorphic principal parts. Exactness of taking stalks givesChoose a holomorphic coordinate $t$ vanishing at $x$. Every meromorphic germ has a finite negative tail in its Laurent series; the holomorphic tail vanishes in the quotient. HenceThis coordinate description is a vector-space identification; the intrinsic space is the quotient of germs, so a coordinate change need not preserve the displayed basis.

For each point inclusion $i_x:\{x\}\to X$, a principal part defines a local meromorphic germ near $x$, whose quotient class is zero off $x$. Gluing with zero on the complement gives a map from its skyscraper sheaf to $\mathcal{P}$. These maps induce an isomorphism on every stalk and therefore an isomorphism of sheaves:The direct sum of sheaves means the sheafification of the sectionwise direct-sum presheaf. Its sections can have infinitely many nonzero components globally, but their point supports form a locally finite family of subsets. This matches the fact that poles of a meromorphic function are locally finite. On a compact Riemann surface only finitely many points can occur; on a noncompact one a discrete infinite family is allowed.

For a global principal-part family $P$, choose local meromorphic lifts $m_i$ on a sufficiently small open cover. The differences $m_j-m_i$ are holomorphic functions and form a Čech cocycle. The connecting homomorphism sends $P$ to the resulting class in $H^1(X,{\mathcal{O}}_X)$. If this class vanishes, after refining the cover write $m_j-m_i=h_j-h_i$; then the functions $m_i-h_i$ glue to a global meromorphic function. Conversely any global lift makes the class zero. Equivalently, exactness of the given long exact sequence in sheaf cohomology saysThe connecting class is the obstruction to the Mittag-Leffler problem on a Riemann surface. It rules on the specified poles and their finite negative Laurent tails, with no additional poles allowed. When a solution exists, any two solutions differ by a global holomorphic function.

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.

Evaluation is surjective as a sheaf morphism: near $p$, any prescribed value is supplied by a constant holomorphic function. We therefore have a short exact sequence of sheaveswhere ${\mathcal{I}}_p$ is the point ideal sheaf and ${\mathcal{O}}_X$ the sheaf of holomorphic functions. By part (c), the skyscraper sheaf is flasque, so $H^0(X,{\mathbb{C}}_p)=\mathbb{C}$ and $H^i(X,{\mathbb{C}}_p)=0$ for $i>0$.

On the connected compact elliptic curve, the maximum modulus principle makes every global holomorphic function constant. Consequently $H^0(X,{\mathcal{O}}_X)=\mathbb{C}$, and evaluation on global sections is an isomorphism. Also $H^1(X,{\mathcal{O}}_X)\cong \mathbb{C}$: by Serre duality for compact complex manifolds, its dual is $H^0(X,K_X)$, generated by the nowhere-zero form $dz$ descended from $\mathbb{C}$. Higher cohomology of ${\mathcal{O}}_X$ vanishes by the Dolbeault theorem in complex dimension one.

The long exact sequence in sheaf cohomology thus beginsIt follows that

On the standard charts $D_{+}(x)$ and $D_{+}(y)$ of ${\mathbb{P}}_k^1$, a section of the twisting sheaf on projective space $\mathcal{O}(d)$ is represented by a degree-zero element of the corresponding localization of $k[x,y](d)$. A global section is therefore a homogeneous polynomial of degree $d$ when $d\ge 0$. There are no nonzero global sections for $d<0$. Hence

Local isomorphism on every member of a fixed affine cover does not imply a global isomorphism: the local identifications may have different transition functions. For example, $\mathcal{O}$ and $\mathcal{O}(1)$ are both trivial on the two standard affine charts of ${\mathbb{P}}^1$, but they are not isomorphic because their spaces of global sections have dimensions one and two.

An injective map between line bundles need not be an isomorphism. Multiplication by a nonzero section giveson ${\mathbb{P}}^1$; its cokernel is a nonzero skyscraper sheaf supported at the zero of the section.

Take $X=\mathbb{R}$, $U=(0,1)$, and let $\mathcal{F}$ be the skyscraper sheaf with value a nonzero abelian group $A$ at $p=3/4$. Cover $V=(-1,1)$ by $V_1=(-1,2/3)$ and $V_2=(1/2,1)$. The presheaf gives $(j^p\mathcal{F})(V)=0=(j^p\mathcal{F})(V_1)$ and $(j^p\mathcal{F})(V_2)=A$. A nonzero section on $V_2$ and the zero section on $V_1$ agree on the overlap, whose skyscraper sections vanish, but cannot be glued on $V$. Thus $j^p\mathcal{F}$ need not be a sheaf.

Sheafification preserves stalks. If $P\in U$, neighborhoods contained in $U$ are cofinal, so $(j^p\mathcal{F}{)}_P={\mathcal{F}}_P$. If $P\notin U$, no neighborhood of $P$ lies in $U$, so every term defining the presheaf stalk is zero. Therefore

For any point inclusion $i:\{x\}\hookrightarrow X$, the skyscraper sheaf $i_{\ast }A$ has restriction maps that are identities or maps to zero. It is flasque, so its higher sheaf cohomology vanishes and its global sections are $A$. This requires no assumption that $x$ is closed.