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Let $X$ be a separated Noetherian scheme over a field $k$ and $\mathcal{F}$ a coherent sheaf. Coherent cohomology studies the groups $H^q(X,\mathcal{F})$: global sections in degree zero and the obstructions to gluing local sections in higher degrees. Coherence means finite generation locally, and on a Noetherian scheme it is preserved by kernels, cokernels and extensions. This makes coherent sheaves suitable for exact-sequence calculations.

For an ordered open cover $\mathcal{U}=(U_i)$, the Čech cochain groups arewith differentialEach double omission occurs twice with opposite signs, so ${\delta }^2=0$. The Čech cohomology of the cover is ${\check{H}}^q=\ker \delta /\mathrm{im}\delta$, and taking the direct limit over refinements defines cover-independent Čech cohomology. In degree zero the sheaf gluing axiom identifies it with $\Gamma (X,\mathcal{F})$.

The comparison with derived-functor sheaf cohomology is particularly effective algebraically. Positive-degree cohomology of a quasi-coherent sheaf on an affine scheme vanishes. A separated scheme has affine intersections of affine opens, so a finite affine cover is acyclic for $\mathcal{F}$. Its Čech cochain complex consequently computes $H^q(X,\mathcal{F})$ directly. For a short exact sequence of sheaves, sections over each affine intersection are exact on quasi-coherent sheaves. The resulting short exact sequence of Čech complexes gives the long exact sequence in sheaf cohomology. In particular, the connecting map measures the failure of a global section of the quotient to lift globally.

Several central projective results make the theory finite and computable. On a projective variety over $k$, every coherent cohomology group is finite-dimensional. The Grothendieck vanishing theorem gives $H^q(X,\mathcal{F})=0$ for $q>\dim X$. If ${\mathcal{O}}_X(1)$ is ample, Serre vanishing gives $H^{q>0}(X,\mathcal{F}(m))=0$ for sufficiently large $m$, and these large twists are globally generated sheaves. On projective space a coherent sheaf has a finite twisting resolution of a coherent sheaf on projective space: finite sums of $\mathcal{O}(a)$ resolve it. The explicit line-bundle cohomology below and the long exact sequences prove finiteness and positive-twist vanishing. For a closed embedding $i:X\hookrightarrow {\mathbb{P}}^n$, $i_{\ast }\mathcal{F}$ is coherent and $H^q(X,\mathcal{F})=H^q({\mathbb{P}}^n,i_{\ast }\mathcal{F})$, reducing these results to projective space. A sufficiently positive twist of a finite presentation by sums of $\mathcal{O}(-a)$ also proves global generation.

The Euler characteristic of a coherent sheafis therefore defined, and the long exact sequence makes it additive. With a fixed projective embedding, $\chi (X,\mathcal{F}(m))$ is its Hilbert polynomial, and for sufficiently large $m$ it equals $h^0(X,\mathcal{F}(m))$. On a smooth projective variety of dimension $r$, Serre duality gives, for a vector bundle $E$,For an arbitrary coherent sheaf the correct counterpart is $H^q(X,\mathcal{F}{)}^{\ast }\cong {\mathrm{Ext}}_X^{r-q}(\mathcal{F},{\omega }_X)$; replacing this Ext group indiscriminately by the cohomology of the ordinary sheaf dual would lose local extension information.

We now calculate the basic examples. For $n\ge 1$, use the standard cover $U_i=D_{+}(x_i)$ of ${\mathbb{P}}^n$. Sections of $\mathcal{O}(m)$ on an intersection are degree-$m$ Laurent monomials in which negative exponents are allowed only for coordinates inverted on that intersection. Decompose the Čech cochain complex by exponent vectors $(a_0,\dots ,a_n)$ with sum $m$. If no exponent is negative, the corresponding simplex complex contributes only in degree zero. If all exponents are negative, the monomial occurs only on the full intersection and contributes only in degree $n$. When the negative-index set is nonempty and proper, inserting any index outside it gives a contracting homotopy, so the monomial complex is acyclic. Hence the cohomology of twisting sheaves on projective space isIn the top group a basis consists of the Laurent monomials with every exponent at most $-1$. Writing $a_i=-1-b_i$ reduces the count to nonnegative solutions of $\sum b_i=-m-n-1$, giving the displayed binomial coefficient. In particular, all intermediate cohomology vanishes, and ${\omega }_{{\mathbb{P}}^n}=\mathcal{O}(-n-1)$ agrees with Serre duality. For ${\mathbb{P}}^0$ separately, every twist is trivial and $H^0=k$.

On the projective line this givesThus positive and negative degrees produce quite different groups, while their alternating difference is a single polynomial. The twisted cubic is isomorphic to ${\mathbb{P}}^1$ and its hyperplane bundle corresponds to ${\mathcal{O}}_{{\mathbb{P}}^1}(3)$. Consequently $h^0(X,{\mathcal{O}}_X(m))=3m+1$ for $m\ge 0$, agreeing with Q2, and $h^1(X,{\mathcal{O}}_X(m))=\max (-3m-1,0)$.

For a smooth plane curve $C$ of degree $d\ge 1$, the ideal-sheaf sequence of a projective hypersurface isThe long exact sequence in sheaf cohomology and the preceding formula give $H^0(C,{\mathcal{O}}_C)=k$ and $H^1(C,{\mathcal{O}}_C)\cong H^2({\mathbb{P}}^2,\mathcal{O}(-d))$. ThereforeFor $d=1,2$ this is zero; for a plane cubic it is one. Twisting the same exact sequence gives $\chi (C,{\mathcal{O}}_C(m))=dm+1-g$. These examples illustrate how Čech cohomology, vanishing, Serre duality and exact sequences convert local equations into global numerical invariants.

A quasi-coherent sheaf is a sheaf of modules which locally has a presentation by free module sheaves:where the index sets may be infinite. Equivalently, on affine neighbourhoods it is an affine module sheaf. A coherent sheaf is locally finitely generated, with the additional requirement that the kernel of every map from a finite free module sheaf is locally finitely generated. On a Noetherian scheme, these conditions are equivalent to being locally associated with a finitely generated module.

For an affine variety $X$ with coordinate ring $A$ and an $A$-module $M$, setRestriction on principal open subsets is the natural module localization map. On an arbitrary open subset, sections are families of these stalk values locally represented by a single fraction from a localization. This constructs the affine module sheaf; its global sections are $M$. Choose a free-module presentation $A^{(J)}\to A^{(I)}\to M\to 0$. Exactness of localization and its compatibility with direct sums yieldThus $\tilde{M}$ is a quasi-coherent sheaf, without a finite-generation assumption.

We now prove coherence of an affine module sheaf. If $M$ is finitely generated, take a surjection $A^r\to M$. Since $A$ is a Noetherian ring, its kernel is finitely generated, so $M$ has a finite presentation of a module. More generally, on every principal affine open, the kernel of a map $A_a^s\to M_a$ is a submodule of a finite free module over the Noetherian ring $A_a$, and is finitely generated. On sheafifying, this verifies both parts of the definition of a coherent sheaf.

Conversely, if $\tilde{M}$ is coherent, choose a finite principal-open cover $X={\bigcup }_{i}D(a_i)$ on which it has finitely many generators. Write those local generators as $m_{ij}/a_i^{r_{ij}}$. Their numerators generate a finite submodule $N={\sum }_{i,j}A{m}_{ij}$ of $M$, and $N_{a_i}=M_{a_i}$ for every $i$. To check that $M/N=0$, let $q\in M/N$. Each equality after localization gives an exponent $r_i$ such that $a_i^{r_i}q=0$. The ideal generated by the $a_i^{r_i}$ is the unit ideal: otherwise a maximal ideal containing all of them would give a point outside the principal-open cover. A linear combination of these powers equal to $1$ consequently annihilates $q$. Every $q$ is zero, proving

Next consider a morphism $\varphi :{\mathcal{G}}_1\to {\mathcal{G}}_2$ between quasi-coherent sheaves. On an affine open $V$ of the target variety, use the assumed affine module-sheaf equivalence to write ${\mathcal{G}}_j{|}_V=\widetilde{M_j}$, with $M_j=\Gamma (V,{\mathcal{G}}_j)$. The sheaf morphism is the sheafification of the module map $\psi =\Gamma (V,\varphi )$. Indeed, its action on a section $m/a^r$ of a principal open is forced by module-linearity and restriction to be $\psi (m)/a^r$. Therefore exactness of localization givesThese identities are compatible with restriction, so $(\ker \varphi ){|}_V=\widetilde{\ker \psi }$. Since this holds on an affine cover, the kernel sheaf is quasi-coherent on every variety. This proof of kernels of quasi-coherent sheaf morphisms does not assert that taking global sections is right exact.

For direct image along a morphism of affine varieties, write $Y$ with coordinate ring $A$, $X$ with coordinate ring $B$, and let $\alpha :A\to B$ be the induced ring homomorphism. If $\mathcal{F}=\tilde{M}$ on $X$, regard $M$ as an $A$-module by restriction of scalars. For $a\in A$,The last equality identifies localization with the scalar action of $a$. It identifies the direct image sheaf with the affine module sheaf of the restricted module, so $f_{\ast }\mathcal{F}$ is quasi-coherent.

Suppose $Y$ is affine and $X={\bigcup }_{i=1}^r{U}_i$ is a finite affine cover. Since a variety is separated, every $U_i\cap U_j$ is affine: it is identified with the inverse image of the closed diagonal morphism inside the affine product $U_i{\times }_k{U}_j$. Write $g_i=f{|}_{U_i}$ and $g_{ij}=f{|}_{U_i\cap U_j}$. Form the morphism of sheaves of modulesBoth sums are finite, and every summand is quasi-coherent by the affine calculation. For every open $V\subseteq Y$, the sheaf gluing axiom on $f^{-1}(V)$ identifies the kernel of this map on sections with $\mathcal{F}(f^{-1}(V))$. HenceThe kernel result proved above establishes quasi-coherence.

Finally, for arbitrary $Y$, take an affine neighbourhood $V\subseteq Y$ and apply the preceding argument to $f^{-1}(V)\to V$. The open subset $f^{-1}(V)$ is again a variety and has a finite affine cover, because varieties have Noetherian Zariski topology. The restriction of $f_{\ast }\mathcal{F}$ to $V$ is exactly the direct image for this restricted morphism. ThusThe finiteness and separation conditions are the relevant hypotheses, not affineness of the original source or target. More generally, quasi-coherence of direct image under a quasi-compact quasi-separated morphism is valid. If one permits nonseparated finite-type prevarieties, each overlap in the argument has a finite affine cover; replacing its summand by the sum of direct images from these charts gives the same kernel, since agreement can be tested on a cover. This also proves the assertion in that convention. It does not extend by this argument to arbitrary spaces with an unrestricted infinite cover.

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.

Choose an ordered affine open cover $\mathcal{U}=(U_i)$ of the separated scheme. For a quasi-coherent sheaf $\mathcal{F}$, the Čech cochain groups and differential areTerms corresponding to empty intersections are zero. Restriction commutes with restriction, so the terms in $d^2$ cancel in pairs. DefineHere an affine cover computes the Čech groups independently of the chosen affine cover. The reason is that finite intersections of affine opens in a separated scheme are affine: the intersection of two is the inverse image of the closed diagonal inside their affine product. Repeating gives the finite-intersection assertion. Quasi-coherent sheaves have no higher cohomology on these intersections. Apply an injective resolution of the sheaf and form the double complex of its Čech cochains. In one direction the higher cohomology vanishes on every affine intersection, leaving precisely the displayed Čech complex. In the other direction the augmented Čech complex of the flasque resolution terms is exact, leaving the global-section resolution computing sheaf cohomology. Thus the two cohomologies agree, and refinement maps between affine-cover complexes induce the same groups. No such affine-intersection claim is being made for arbitrary nonseparated schemes.

For an exact sequence of quasi-coherent sheaves, sections on an affine intersection give an exact sequence of modules, including surjectivity: this follows from the equivalence between quasi-coherent sheaves and modules on an affine scheme. Taking products yields a short exact sequence of Čech cochain complexes. To describe the connecting map, lift a degree-$p$ cocycle in $C^p({\mathcal{F}}^{\prime \prime })$ to a cochain in $C^p(\mathcal{F})$. Its differential maps to zero in $C^{p+1}({\mathcal{F}}^{\prime \prime })$ and therefore is a cocycle in $C^{p+1}({\mathcal{F}}^{\prime })$. Changing the lift adds a coboundary there; changing the original representative does the same. This defines the connecting homomorphism. The same lift-and-differentiate argument shows that a class is in its kernel exactly when it lifts to a cocycle in the middle complex, and verifies exactness at the other terms. Thus the resulting sequence isThe sequence must be in quasi-coherent sheaves for this affine-sections argument; that is the intended category here.

For the punctured affine plane, use $U=D(x)$ and $V=D(y)$. Their rings are $k[x^{\pm 1},y]$ and $k[x,y^{\pm 1}]$, and the intersection ring is $k[x^{\pm 1},y^{\pm 1}]$. The two-open Čech complex isConsequentlyThe denominator contains precisely the span of Laurent monomials for which at least one exponent is nonnegative. Every Laurent polynomial has a unique monomial expansion, so the remaining monomials are linearly independent in the quotient and span it. Thereforea countably infinite-dimensional vector space. The brackets here denote cohomology classes. This calculates the Čech cohomology of the punctured affine plane, rather than merely observing that some nonzero cohomology exists.

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:

Work on an affine open subset $U=\mathrm{Spec}A$ and write $\mathcal{F}{|}_U=\tilde{M}$ and $\mathcal{E}{|}_U=\tilde{N}$ using part (ii). We first show that any section $e\in \Gamma (U,\mathcal{E})$ lifts to $\Gamma (U,\mathcal{G})$.

A surjection of sheaves supplies local lifts $g_i$ on a finite cover by principal open subsets $D(s_i)$. Their differencessatisfy $c_{ij}+c_{jk}=c_{ik}$ on triple intersections. We spell out why this Čech cocycle is a coboundary. Choose a common sufficiently large power $n$ so that $s_j^n{c}_{ij}$ can be represented in $M_{s_i}$ for every pair. Increasing $n$ also kills the finitely many localization errors in the cocycle identities: after extending $s_k^n{c}_{ik}$ to $M_{s_i}$ and $s_k^n{c}_{jk}$ to $M_{s_j}$, their difference in $M_{s_i{s}_j}$ is $s_k^n{c}_{ij}$. This follows because any error which vanishes after localizing at $s_k$ is killed by a power of $s_k$; there are only finitely many indices.

The $D(s_i^n)$ still cover $U$, so the ideal generated by the $s_i^n$ is $A$. Choose $a_i\in A$ with ${\sum }_i{a}_i{s}_i^n=1$ and putusing the extended representatives just chosen. On an overlap, the cocycle identity givesThus $g_i-b_i$ agree and glue to a lift of $e$. This is the degree-one argument behind the exactness of the unit-ideal localization Čech complex.

The same reasoning applies to every $D(a)\subseteq U$. Hence we have short exact sequences of modules of sections on $U$ and on $D(a)$. Localize the first sequence at $a$ and compare them. The maps on the outer modules are the isomorphisms from part (ii):An elementary diagram chase makes the middle map an isomorphism: subtract a lift from the right to reduce surjectivity to the left, and use injectivity on the left to prove injectivity. These isomorphisms identify $\mathcal{G}{|}_U$ with the quasi-coherent sheaf associated with $\Gamma (U,\mathcal{G})$. Doing this on every affine open subset provesThe proof even works without the Noetherian assumption.

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,

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.

Put $P={\mathbb{P}}_k^2$ and $S=k[t_0,t_1,t_2]$. A homogeneous polynomial $F$ of positive degree $d$ is nonzero, and $S$ is an integral domain. Multiplication by $F$ therefore gives an exact sequence of coherent sheaveswhere $i:X\hookrightarrow P$ is the closed immersion.

We compute the necessary cohomology of twisting sheaves on projective space. The standard three affine open subsets $D_{+}(t_i)$ and all their intersections are affine schemes. For a quasi-coherent sheaf, vanishing of quasi-coherent cohomology on an affine scheme and the acyclic cover theorem identify its sheaf cohomology with the Čech cohomology of this cover. In degree $m$, the Čech cochain complex isDecompose it into subcomplexes indexed by Laurent monomials $t_0^{a_0}{t}_1^{a_1}{t}_2^{a_2}$ with $a_0+a_1+a_2=m$. Let $N=\{i:a_i<0\}$. Such a monomial occurs in the summand for a nonempty index set $J$ precisely when $N\subseteq J$, since only the variables in $J$ are inverted.

If $N$ is empty, the monomial contributes the cochain complex of a full two-simplex: its degree-zero cohomology is $k$ and its higher cohomology vanishes. If $|N|=1$, the dimensions in degrees $0,1,2$ are $1,2,1$; the first map has rank one and the second map has rank one with the same kernel as the first map's image. If $|N|=2$, there is one copy of $k$ in each of degrees $1$ and $2$ and the map between them is an isomorphism. These two cases are acyclic over every field, including characteristic two. If $N=\{0,1,2\}$, only the degree-two term occurs, contributing one copy of $k$. It follows thatand that $H^0(P,{\mathcal{O}}_P)=k$, $H^0(P,{\mathcal{O}}_P(-d))=0$, and $H^2(P,{\mathcal{O}}_P)=0$. Moreover the Laurent-monomial description of top cohomology on projective space gives a basis of $H^2(P,{\mathcal{O}}_P(-d))$ consisting ofFor $d\ge 3$, subtracting one from each $b_i$ counts these as $\left(\genfrac{}{}{0ex}{}{d-1}{2}\right)$. For $d=1,2$ there are no such triples, and the formula $(d-1)(d-2)/2$ is also zero.

Use the long exact sequence in sheaf cohomology of the initial sequence, and cohomology under a closed immersion to identify the terms for $i_{\ast }{\mathcal{O}}_X$. The vanishings just proved giveThereforeThis proves the cohomology of a projective plane hypersurface without imposing smoothness, irreducibility or reducedness. In particular, the second number is an arithmetic genus, not a claim about the genus of a normalization.

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.