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For an affine variety with coordinate ring $A$ and an $A$-module $M$, localization defines sections on principal open subsets by the displayed formula. Restrictions are the natural localization maps. Equivalently, a section is a family of stalk values which locally come from one fraction $m/f^r$. These data form a quasi-coherent sheaf, with global sections $M$ and stalk $M_{{\mathfrak{m}}_P}$ at $P$. Sheaf tensor products correspond to tensor products over $A$; pullback of a sheaf of modules corresponds to extension of scalars, and affine direct image to restriction of scalars.

Let $X$ be an algebraic variety over an algebraically closed field $k$, and assume $H^1(X,\mathcal{I})=0$ for every coherent ideal sheaf $\mathcal{I}$. Given a closed point $P$, choose an affine open subscheme $U$ containing it and give $Y=X\setminus U$ its reduced induced subscheme structure. Since $P\notin Y$, evaluation at $P$ is a surjective morphism of sheaveswith kernel ${\mathcal{I}}_Y\cap {\mathcal{I}}_P$. Here $k_P$ is the skyscraper sheaf at the closed rational point, and the kernel is again a coherent ideal sheaf. The long exact sequence in sheaf cohomology therefore lifts $1\in k_P$ to a global section $f$ of ${\mathcal{I}}_Y$.

Viewed as a global regular function, $f$ vanishes on $Y$ and has $f(P)=1$. Thus $P\in X_f\subseteq U$, and $X_f=U_{f{|}_U}$ is a principal open subscheme of an affine scheme, hence affine. Such neighborhoods cover $X$: every nonempty closed subset of an algebraic variety contains a closed point. This is the local step in the cohomological criterion for affineness; the closed-point hypothesis makes the coherent skyscraper argument valid.

For an affine scheme $X=\mathrm{Spec}A$, its ring of global sections is $A$. A morphism of schemes between affine schemes is determined contravariantly by its map on these rings. Consequently a morphism between affine schemes is an isomorphism of schemes exactly when that ring map is an isomorphism. The affine hypothesis is essential, as the punctured affine plane shows.

A morphism of schemes into an affine scheme is uniquely determined by its homomorphism on global sections. Conversely, a ring homomorphism $A\to \Gamma (X,{\mathcal{O}}_X)$ restricts on every affine open subscheme of $X$ to a map into $\mathrm{Spec}A$; equality on affine covers of overlaps glues these maps. No quasi-compactness or affineness assumption on $X$ is needed.

On a quasi-compact scheme $X$, a line bundle $L$ is ample if each point has an affine neighborhood of the formEquivalently, the affine nonvanishing loci of global sections of positive tensor powers form a basis of a topology. This makes ampleness a condition that global sections recover enough affine local geometry.^[ref]

For a proper scheme over a field, this is equivalent to some positive tensor power being very ample: $L^{\otimes m}$ is the pullback of $\mathcal{O}(1)$ by a closed immersion into projective space. A positive tensor power may be needed. On an elliptic curve, a degree-one line bundle is ample but has only one independent global section, so it cannot itself embed the curve. Its third power has degree three and gives a plane cubic embedding. On an affine scheme, every line bundle is ample under the scheme definition, even when the scheme is not proper.

For a Noetherian ring $A$, a finitely generated module $M$ has a finite presentation of a module, and kernels of maps from finite free modules to $M$ are finitely generated. Its affine module sheaf is therefore a coherent sheaf. Conversely, if $\tilde{M}$ is coherent, choose a finite principal open subset cover on which its section modules are finitely generated. Lift their finitely many generators to numerators in $M$. The submodule generated by these numerators becomes all of $M$ on every chart, and hence is all of $M$: its quotient has zero localization at every point. This proves finite generation without assuming that global sections preserve arbitrary sheaf surjections.

A Noetherian scheme is affine exactly when every coherent ideal sheaf has zero first sheaf cohomology. One direction is vanishing of quasi-coherent cohomology on an affine scheme. For the converse, ideal-sheaf vanishing produces affine principal neighbourhoods from ideal-sheaf vanishing. A finite such cover yields a unit-ideal certificate from a principal affine cover, and those affine charts glue to the spectrum of a commutative ring of global sections. The criterion, including the finite-type ideal version for quasi-compact quasi-separated schemes, is recorded in Stacks Project, Section 30.3.

For a morphism of sheaves $\varphi :\mathcal{F}\to \mathcal{G}$ of abelian groups or modules, the cokernel sheaf is the sheafification of the presheaf $U\mapsto \mathcal{G}(U)/\varphi (\mathcal{F}(U))$. Its stalk at $x$ is $\mathrm{coker}({\mathcal{F}}_x\to {\mathcal{G}}_x)$. The presheaf quotient itself need not satisfy the sheaf gluing axiom, which explains why a surjective morphism of sheaves need not be surjective on global sections.

The quotient sheaf ${\mathcal{K}}_X^{\ast }/{\mathcal{O}}_X^{\ast }$ records the zero and pole orders of local nonzero meromorphic functions. Each of its global sections is naturally a divisor on a complex manifold on $X$.

For the sheaf ${\Omega }^p$ of holomorphic $p$-forms on a complex manifold, the Dolbeault-Poincaré lemma makesa resolution by fine sheaves. Taking global sections therefore gives the natural isomorphism

Work over an algebraically closed field of characteristic different from two. A smooth projective plane curve $B\subset {\mathbb{P}}^2$ of degree $2d$ is the zero set of a global section $s$ of $L^{\otimes 2}$, where $L={\mathcal{O}}_{{\mathbb{P}}^2}(d)$. The finite morphism $\pi :X\to {\mathbb{P}}^2$ defined locally by $w^2=s$ is a double cover, branched along $B$. The Jacobian criterion shows that $X$ is smooth: away from $B$ the derivative in $w$ is nonzero, while on $B$ some derivative of $s$ is nonzero.

The ramification divisor $R$ satisfies ${\mathcal{O}}_X(R)={\pi }^{\ast }L$. The determinant of the differential giveswhere $H$ is the hyperplane class. Thus $K_X$ is ample for $d\ge 4$, so $X$ is a surface of general type; its self-intersection number is $K_X^2=2(d-3{)}^2$. A smooth octic branch curve gives $K_X^2=2$. For $d=3$, the same formula instead gives a trivial canonical line bundle of a smooth variety, illustrating why the branch degree changes the classification. The assumption on characteristic is needed for this local double-cover construction and its smoothness argument.

An effective Cartier divisor is cut out locally by one non-zero-divisor; it is a codimension-one closed subscheme. On an integral scheme a nonzero global section of a line bundle defines such a divisor, possibly empty. On a nonreduced scheme a nonzero section need not be regular, so its zero locus need not be a Cartier divisor.

On ${\mathbb{P}}_k^n$ with $n>0$, let $\mathcal{F}=i_{\ast }k$ at a rational point. It has one-dimensional global sections, but its dual of a sheaf is zero: a local map from the residue field into the local polynomial ring must have image annihilated by a nonzero parameter, which is impossible in an integral domain. Thus $H^n({\mathcal{F}}^{\vee }(-n-1))=0$. Serre duality for arbitrary coherent sheaves instead uses an Ext functor; replacing this by the ordinary dual can lose torsion information.

A globally generated line bundle on a quasi-compact scheme is generated by finitely many global sections. Each point has a section generating on an open neighbourhood, and quasi-compactness gives a finite subcover. This supplies a map into a fixed finite-dimensional projective space associated with that line bundle.

A first-order embedded deformation of a closed subscheme $Z\subseteq X$ is a closed subscheme $Z_{\epsilon }\subseteq X{\times }_k\mathrm{Spec}(k[\epsilon ]/({\epsilon }^2))$ flat over the dual numbers and with special fibre equal to the given embedded $Z$. It is a tangent vector of the Hilbert scheme, not a deformation modulo arbitrary automorphisms of the ambient scheme. These deformations are naturally the global sections of the normal sheaf.

The embedding line bundle has at least four independent global sections. If it is nonspecial, the Riemann-Roch theorem gives $h^0=6-g$, so $g\le 2$. If it is special, the Clifford inequality for curves gives $h^0\le 3$, a contradiction. The smooth projective curve need not span its full ambient projective space.

For an ample line bundle $L$ on a noetherian scheme $X$, every coherent sheaf $F$ becomes a globally generated sheaf after tensoring with $L^{\otimes n}$ for all sufficiently large $n$. Finitely many global sections suffice by finite global generation on a quasi-compact scheme. This is the global-generation characterization of an ample line bundle.^[ref]

For a closed subscheme of projective space, take $L={\iota }^{\ast }\mathcal{O}(1)$. One way to see why twisting helps is to choose finitely many module generators on each standard affine open subscheme. Multiplication by a sufficiently high power of its homogeneous coordinate extends each generator to a global section of a twist. Choose one exponent for all generators. On the original chart that coordinate is invertible, so these extended sections still generate; doing this for every chart proves global generation. Tensoring back by the inverse invertible sheaf gives a surjection from a finite direct sum of sheaves of negative twists onto $F$.

A line bundle is globally generated if at each point some global section has a germ generating its stalk as a module over the local ring. Equivalently its evaluation map from global sections tensored with the structure sheaf is surjective.

The displayed line bundle is globally generated, but no finite family of global sections generates it. A family of $N$ sections restricts to $N$ linear forms on ${\mathbb{P}}^m$; for $m\ge N$, these have a common zero. This shows why finite global generation on a quasi-compact scheme requires its finiteness hypothesis.

A sheaf of modules is globally generated when the evaluation map from its global sections is surjective. Equivalently, the germs of global sections span each stalk as a module over the local ring. For a coherent sheaf on a projective variety, every sufficiently positive twist is globally generated: embed the variety in projective space, present its direct-image sheaf as a quotient of a finite sum of $\mathcal{O}(-a)$, twist until those line bundles are globally generated, and restrict their generating sections to the variety.