Ringed space

A ringed space $(X,{\mathcal{O}}_X)$ is a topological space $X$ equipped with a sheaf of rings ${\mathcal{O}}_X$. A morphism consists of a continuous map $f:X\to Y$ and a compatible morphism $f^{-1}{\mathcal{O}}_Y\to {\mathcal{O}}_X$.

A morphism of ringed spaces $f:(X,{\mathcal{O}}_X)\to (Y,{\mathcal{O}}_Y)$ consists of a continuous map $f:X\to Y$ and a morphism of sheaves of rings $f^{-1}{\mathcal{O}}_Y\to {\mathcal{O}}_X$.

A sheaf of rings assigns a ring $\mathcal{O}(U)$ to every open set, restriction homomorphisms to inclusions, and satisfies local identity and gluing. It supplies the local functions on a ringed space.

This is a ring-valued Sheaf (mathematics), with ring homomorphisms as restriction maps.

A sheaf of topological rings takes values in topological rings, with continuous restriction maps. For an open cover, the sheaf condition is an equalizer of the product of sections on the cover and the product on pairwise overlaps, in the category of topological rings. The structure sheaf of a formal scheme is an inverse limit of discrete structure sheaves with this topology.

// Target: algebra.bigb

For a ringed space $(X,{\mathcal{O}}_X)$, a sheaf of ${\mathcal{O}}_X$-modules is a sheaf $\mathcal{F}$ such that every $\mathcal{F}(U)$ is an ${\mathcal{O}}_X(U)$-module and restriction maps preserve scalar multiplication.

The underlying object is a Sheaf (mathematics) with compatible local module actions.

For sheaves of modules $\mathcal{F},\mathcal{G}$ on a ringed space, the internal Hom sheaf assigns to an open $U$ the module of sheaf morphisms $\mathcal{F}{|}_U\to \mathcal{G}{|}_U$ respecting ${\mathcal{O}}_U$. Compatible morphisms glue, so this is a sheaf. Its global sections are precisely the global module-sheaf morphisms. For a locally free sheaf $\mathcal{F}$ of finite rank it is ${\mathcal{F}}^{\vee }\otimes \mathcal{G}$.

A local system is a locally constant sheaf of modules, for example the integral cohomology of fibres of a smooth proper family of compact complex manifolds. Parallel continuation around loops gives its monodromy representation.

A rational section is a section on a dense open subset, with two representatives identified when they agree on a common dense open subset. Intersecting representatives' domains defines addition. Multiplication by a similarly represented rational function on an algebraic variety makes these classes a module over the ring $\mathrm{Rat}(X)$. For an irreducible variety, this ring is its function field.

For a locally free sheaf on an irreducible variety, the stalk inclusion identifies sections regular near $P$ inside its rational sections. The quotient records the obstruction to regularity at that point. This definition does not assume a nonsingular point or a discrete valuation parameter.

On an irreducible algebraic curve, let $\mathcal{R}(\mathcal{F})$ be the constant rational-section sheaf and put $\mathcal{P}(\mathcal{F})(U)={⨁}_{P\in U}\mathrm{Rat}(\mathcal{F})/{\mathcal{F}}_P$. Every open subset is quasi-compact, so compatible finite-support families glue with finite support. Its stalk at $P$ is the displayed rational-principal-part quotient. A rational section is regular outside finitely many points, making its principal-part map well defined. The resulting sequence is stalkwise exact. Both right-hand sheaves are flasque, and consequently $H^1(X,\mathcal{F})$ is the cokernel of the global principal-part map and $H^q(X,\mathcal{F})=0$ for $q\ge 2$. The curve and Zariski-topology hypotheses distinguish finite support here from locally finite pole sets in complex analysis.

Let $X$ be an irreducible affine curve, $A=k[X]$, $K=\mathrm{Frac}A$, and $\mathcal{F}=\tilde{M}$ be locally free. Given finitely many classes in $M_K/M_{{\mathfrak{m}}_P}$, choose one nonzero $a\in A$ clearing denominators of representatives. The ring $A/(a)$ is Artinian and is the product of its finitely many local factors. Hence $M/aM\cong {⨁}_{P\in V(a)}(M/aM{)}_{{\mathfrak{m}}_P}$. Prescribe $a{v}_P$ modulo $a{M}_{{\mathfrak{m}}_P}$ at the requested points and zero at the other zeros of $a$. Lift this tuple to $w\in M$; the section $w/a$ has exactly the requested rational principal parts and no others. This proves surjectivity without assuming the curve is nonsingular.

On an irreducible variety, every nonempty open subset is dense, so a germ determines a rational section. For a locally free sheaf, this map is injective: a local section which vanishes on a dense open has all its regular coefficient functions zero, hence is zero. Local freeness is important; a section of a torsion sheaf can vanish generically without having zero germ.

On a complex manifold, a sheaf of modules over the sheaf of holomorphic functions is coherent analytic if it is locally finitely generated and the kernel of every morphism ${\mathcal{O}}^r\to \mathcal{F}$ is locally finitely generated. The structure sheaf and locally free finite-rank sheaves are examples. Cartan theorem B gives vanishing of their higher sheaf cohomology on a Stein manifold.

The dual is the sheaf of modules whose sections on $U$ are module-sheaf morphisms $\mathcal{F}{|}_U\to {\mathcal{O}}_U$. For a locally free sheaf of finite rank, the dual is locally free of the same rank and the natural double-dual map is an isomorphism. General coherent sheaves need not have this property: a skyscraper at a point of positive-dimensional projective space has zero ordinary sheaf dual.

On a ringed space $(X,{\mathcal{O}}_X)$, an injective module sheaf is an injective object in the category of sheaves of modules over ${\mathcal{O}}_X$. Its defining extension property concerns module-sheaf morphisms; it is distinct from injectivity solely in the category of sheaves of abelian groups.

For opens $U\subseteq V\subseteq X$, there is a monomorphism between the extensions by zero of ${\mathcal{O}}_U$ and ${\mathcal{O}}_V$. Morphisms from these sheaves into an injective sheaf of modules $I$ identify with sections on $U$ and $V$, respectively. The injective object extension property therefore makes the restriction $I(V)\to I(U)$ surjective, proving that $I$ is flasque. The argument applies to any ringed space.

The stalk ${\mathcal{F}}_x$ is the direct limit of $\mathcal{F}(U)$ over neighborhoods $U$ of $x$. A morphism of sheaves is an isomorphism or forms an exact sequence exactly when it does so on every stalk.

The derived costalk at $x$ is $i_x^{!}E$, where $i_x:\{x\}\hookrightarrow X$. It computes local cohomology supported at $x$. On an oriented $N$-manifold the constant sheaf has costalk $\mathbb{Q}[-N]$, whose cohomology is in degree $N$.

For presheaves of sets, $F_x={\underrightarrow{\lim }}_{U\ni x}F(U)$. This functor preserves finite limits because the neighborhood system is filtered and filtered colimits commute with finite limits in sets. Its right adjoint sends a set $S$ to the presheaf taking the value $S$ on opens containing $x$ and the singleton on other opens.

The joint stalk functor $\mathbf{Sh}(X)\to {\mathbf{Set}}^X$ is faithful: equality of germs makes two images of a section locally equal, and the sheaf uniqueness axiom makes them globally equal. Its right adjoint is the product of the set-valued skyscraper sheaves at all points. The Beck comonadicity theorem then makes the joint stalk functor comonadic.

For a point inclusion $i:\{x\}\hookrightarrow X$ and an abelian group $A$, the skyscraper sheaf $i_{\ast }A$ has sections $A$ on opens containing $x$ and zero on other opens. Its restriction maps are identities or maps to zero. If $A\ne 0$, its nonzero stalks occur at every point of the closure $\overline{\{x\}}$: exactly those points whose every neighbourhood contains $x$. Thus the familiar assertion that only the stalk at $x$ is nonzero requires $x$ to be a closed point. It is always flasque, giving the stated skyscraper sheaf cohomology.

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.

The germ $s_x$ of a section $s\in \mathcal{F}(U)$ is its image in the stalk of a sheaf ${\mathcal{F}}_x$. It is zero exactly when $s$ restricts to zero on some neighbourhood of $x$.

The support of a sheaf section is the closed setIts complement is open because a zero germ of a sheaf section is represented by a section that vanishes on a neighbourhood.

Sheafification associates a sheaf to a presheaf without changing its stalks and is universal among morphisms from that presheaf to sheaves.

Every presheaf morphism from $\mathcal{F}$ to a sheaf $\mathcal{H}$ factors uniquely through the canonical sheafification map. Local representatives give local images in $\mathcal{H}$, and the sheaf gluing axiom makes their union unique.

For a presheaf, take families of elements of its stalks which near each point are the germs of one presheaf section. Pointwise operations and restriction make these families into a sheaf. The map sending a section to all its germs is its sheafification map.

For an open inclusion $j:U\hookrightarrow X$, extension by zero is the sheaf $j_{!}\mathcal{F}$ whose stalk equals ${\mathcal{F}}_x$ on $U$ and zero outside $U$. Its sections are local sections on $U$ whose support is closed in the ambient open set.

A sheaf of modules on a scheme is quasi-coherent when every affine chart $\mathrm{Spec}A$ restricts it to the sheaf associated with an $A$-module. On an affine scheme it is determined by its module of global sections.

On an affine scheme, a morphism between quasi-coherent sheaves is determined by the induced map on global sections, because its maps on principal open subsets are their localizations. Exactness of localization gives the displayed identity. Thus the kernel sheaf of a morphism between quasi-coherent sheaves is quasi-coherent on an arbitrary scheme, by applying the affine argument locally. No finite generation is needed.

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.

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.

In $0\to \mathcal{F}\to \mathcal{G}\to \mathcal{E}\to 0$, assume the outer sheaves are quasi-coherent. On an affine open, any section of $\mathcal{E}$ has local lifts to $\mathcal{G}$ on a finite principal cover. Their differences are a Čech cocycle in the module defining $\mathcal{F}$. Exactness of the unit-ideal localization Čech complex makes this cocycle a coboundary, so adjusted lifts glue. Thus sections are exact on the affine open and on every principal open in it. Comparing the resulting sequences after localization, the outer isomorphisms force the middle localization map to be an isomorphism. Hence $\mathcal{G}$ is associated with its global-section module and is quasi-coherent.

Choose a finite cover of an affine scheme by principal opens on which the quasi-coherent sheaf is associated with a module. The sheaf gluing axiom expresses global sections as the kernel of the difference map from the finite product of chart sections to the finite product of overlap sections. Localization commutes with kernels and finite products. Localizing at $a$ therefore gives the same kernel for the cover intersected with $D(a)$, proving the formula. The principal-open formulas identify the sheaf with that associated with its global-section module; no finite-generation hypothesis on the module is needed.

On an affine variety or affine scheme with coordinate ring $A$, the global section functor and $M\mapsto \tilde{M}$ are inverse equivalences between modules and quasi-coherent sheaves. On every principal open subset, sections are the appropriate localization of a module. Localization detects exactness at maximal ideals, so these functors are exact even for modules which are not finitely generated.

On an integral scheme, a quasi-coherent sheaf is torsion-free when its module on every affine chart is a torsion-free module over that chart's integral domain. It embeds into its generic fibre. Therefore a torsion-free sheaf with zero generic fibre is zero; locally free sheaves and their subsheaves are torsion-free.

Every quasi-coherent sheaf on an affine scheme has zero higher sheaf cohomology. Together with the equivalence between modules and quasi-coherent sheaves on an affine scheme, this makes affine varieties the basic acyclic pieces for computing cohomology. An affine open cover of a separated scheme has affine finite intersections, so the acyclic cover theorem identifies its Čech cohomology with sheaf cohomology.

On a locally Noetherian scheme, a coherent sheaf is a quasi-coherent sheaf that is locally associated with a finitely generated module. Kernels, cokernels, and finite direct sums of coherent sheaves remain coherent.

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.

On ${\mathbb{P}}_k^1$, $\mathcal{O}$ and $\mathcal{O}(-1)$ are line bundles, so each stalk of either is a free rank-one module over the local ring. Nevertheless their spaces of global sections are $k$ and zero. Indeed a section of $\mathcal{O}(-1)$ on the two standard charts would require a polynomial $a(t)$ to equal $t^{-1}b(t^{-1})$, which has only strictly negative powers. Only the zero section can satisfy this. Pointwise existence of unrelated stalk isomorphisms does not supply a morphism of sheaves.

An ideal sheaf of a closed subscheme that is a coherent sheaf as a module over the structure sheaf. On a Noetherian scheme, every quasi-coherent ideal sheaf is coherent.

A coherent sheaf on ${\mathbb{P}}_k^n$ is the sheaf associated with a graded module for a finite graded $k[t_0,\dots ,t_n]$-module. Choose a finite graded free resolution by the Hilbert syzygy theorem and sheafify it; exactness of localization preserves exactness. All terms are finite sums of twisting sheaves on projective space. Twisting sufficiently positively makes every term acyclic in positive degree, and dimension shifting in long exact sequences in sheaf cohomology proves Serre vanishing. The graded-module description is supplied by Stacks Project, Section 30.15.

The support dimension is the dimension of a scheme of the closed support of a coherent sheaf. Tensoring with a line bundle leaves its support unchanged. Quotienting by a section that avoids its associated points reduces the support dimension by at least one.

An associated point of a coherent sheaf is a point whose corresponding prime is the annihilator of an element of the module on an affine chart. There are finitely many on a Noetherian scheme. A section of a line bundle avoiding them acts injectively on the sheaf, even when the scheme is nonreduced. Over an infinite field a sufficiently ample linear system supplies such a section.

If $i:Z\hookrightarrow X$ is a closed immersion of Noetherian schemes and $\mathcal{F}$ is a coherent sheaf on $Z$, then $i_{\ast }\mathcal{F}$ is coherent on $X$. Affine-locally this regards a finitely generated $A/I$-module as a finitely generated $A$-module.

The tensor product of two sheaves of ${\mathcal{O}}_X$-modules is the sheafification of $U\mapsto \mathcal{F}(U){\otimes }_{{\mathcal{O}}_X(U)}\mathcal{G}(U)$. Its stalk at $x$ is ${\mathcal{F}}_x{\otimes }_{{\mathcal{O}}_{X,x}}{\mathcal{G}}_x$.

For a continuous map $f:X\to Y$ and a sheaf $\mathcal{F}$ on $X$, the direct image is defined by $(f_{\ast }\mathcal{F})(V)=\mathcal{F}(f^{-1}V)$ for every open $V\subseteq Y$.

The direct image sheaf is the result of the direct image functor: $(f_{\ast }\mathcal{F})(V)=\mathcal{F}(f^{-1}V)$.

For a continuous map $f:X\to Y$, the derived direct image $R{f}_{\ast }$ is obtained by applying $f_{\ast }$ to an injective resolution. Composition with derived global sections gives $R\Gamma (Y,R{f}_{\ast }E)\cong R\Gamma (X,E)$.

For a proper map of locally compact spaces and a constructible complex $E$, the derived stalk satisfies $(R{f}_{\ast }E{)}_y\cong R\Gamma (f^{-1}(y),E{|}_{f^{-1}(y)})$. In particular $(R^i{f}_{\ast }\mathbb{Q}{)}_y\cong H^i(f^{-1}(y);\mathbb{Q})$. Properness is what prevents fibre cohomology from missing nearby sections.

For $f:{\mathbb{P}}_k^1\to \mathrm{Spec}k$, let $D$ consist of $0$ and infinity. Its ideal sheaf is $\mathcal{O}(-2)$, giving an exact coherent sequence $0\to \mathcal{O}(-2)\to \mathcal{O}\to {\mathcal{O}}_D\to 0$. The map on global sections is the diagonal $k\to k\oplus k$, which is not surjective. Since direct image to the point is global sections, projectivity does not make direct image right exact.

For a finite morphism between Noetherian schemes, a quasi-coherent sheaf is coherent exactly when its direct image sheaf is coherent. Over an affine target chart, let $B$ be the finite algebra over the target ring $A$, and let $M$ represent the sheaf on $\mathrm{Spec}B$. Finite generation of $M$ over $A$ implies finite generation over $B$, using the same generators; finite generation over $B$ implies finite generation over $A$ because $B$ is finite over $A$.

For an open inclusion $j:U\hookrightarrow X$, this sheaf has sections $\mathcal{F}(V\cap U)$ on an open set $V$. Restriction defines $\mathcal{F}\to {}_U\mathcal{F}$. Its stalks outside $U$ may be nonzero, so it differs from extension by zero.

Tensoring sections on $f^{-1}V$ gives a balanced pairing of the two direct image sheaves on $V$. The universal property of the tensor product of modules and sheafification produce the displayed morphism. Composing with the pullback-direct-image adjunction unit gives the projection formula for sheaves. The comparison itself need not be an isomorphism.

For a closed inclusion $i:X\hookrightarrow Y$, direct image sheaf formation is exact: stalks on $X$ are unchanged and stalks outside $X$ are zero. It also preserves flasque sheaves. Pushing forward a flasque resolution thus gives a resolution computing cohomology on $Y$, and its complex of global sections is identical to the original one on $X$. This proves the displayed isomorphism in every nonnegative degree.

For an affine morphism $f:X\to Y$, the direct image of every quasi-coherent sheaf on $X$ is quasi-coherent. On affine charts this is the fact that restriction of scalars turns the module sheaf $\tilde{M}$ into the module sheaf associated with the same underlying module over the target ring.

Let $j:{\mathbb{A}}_k^1\hookrightarrow {\mathbb{P}}_k^1$ be a standard affine chart and $\mathcal{G}=\tilde{M}$ a quasi-coherent sheaf there. Its direct image is quasi-coherent because $j$ is an affine morphism. The standard two-chart acyclic cover has Čech cochain complex $M\oplus M_x\to M_x$ in degrees zero and one, with differential $(m,n)\mapsto n-m/1$. Surjectivity gives $H^i({\mathbb{P}}_k^1,j_{\ast }\mathcal{G})=0$ for $i>0$. This does not assert that the direct image itself is flasque.

If $f:X\to Y$ is quasi-compact and quasi-separated, then the direct image of every quasi-coherent sheaf on $X$ is again a quasi-coherent sheaf. A finite affine cover of each inverse image of an affine target chart expresses the direct image as the kernel of a morphism between finite sums of quasi-coherent direct images from affine charts.

The inverse image sheaf $f^{-1}\mathcal{E}$ is the sheaf associated with the presheaf whose sections near $U\subseteq X$ are obtained as a colimit of $\mathcal{E}(V)$ over open sets $V\supseteq f(U)$.

This is the sheaf obtained by applying the inverse image functor.

For sheaves of abelian groups and a continuous map $f:X\to Y$, there is a natural bijection ${\mathrm{Hom}}_X(f^{-1}\mathcal{G},\mathcal{F})\cong {\mathrm{Hom}}_Y(\mathcal{G},f_{\ast }\mathcal{F})$. It comes from the universal property of an inverse image sheaf. The inverse image of abelian sheaves is different from the tensor-adjusted pullback of a sheaf of modules on a ringed space.

Every f-morphism of sheaves from $\mathcal{G}$ to $\mathcal{F}$ factors uniquely through the canonical f-morphism from $\mathcal{G}$ to its inverse image sheaf. A local inverse-image section is represented by sections of $\mathcal{G}$ on image neighbourhoods; their prescribed images glue in $\mathcal{F}$.

For a morphism of ringed spaces, the pullback of an ${\mathcal{O}}_Y$-module is

The inverse image functor supplies the sheaf $f^{-1}\mathcal{E}$; tensoring with the source structure sheaf supplies the module scalar extension.

The restriction meant here is the pullback of a sheaf of modules to the closed subvariety, including tensoring with its structure sheaf. On an affine chart, tensoring $M$ with $A/I$ gives $M/IM$. This distinction matters because restricting only the underlying topological sheaf would retain the ambient ring action.

For a morphism of ringed spaces, the pullback of a sheaf of modules is left adjoint to the direct image sheaf functor. The unit sends a local section $h$ to $1\otimes h$ after inverse image. On the structure sheaf it becomes the given ring homomorphism ${\mathcal{O}}_Y\to f_{\ast }{\mathcal{O}}_X$.

For a morphism of ringed spaces, there is a natural morphismIt is an isomorphism when $\mathcal{E}$ is a locally free sheaf of finite rank, because the claim is local and then reduces to distributivity over a finite direct sum.

The same formula holds for higher direct images: $R^i{f}_{\ast }\mathcal{F}\otimes \mathcal{E}\cong R^i{f}_{\ast }(\mathcal{F}\otimes f^{\ast }\mathcal{E})$ when $\mathcal{E}$ is locally free of finite rank.

A sheaf of ${\mathcal{O}}_X$-modules is locally free of finite rank when every point has an open neighborhood $U$ on which it is isomorphic to ${\mathcal{O}}_U^{\oplus r}$ for some finite $r$. If $X$ is connected, the rank is constant.

A sheaf of modules is locally free of rank $r$ if each point has a neighborhood $U$ on which it is isomorphic to ${\mathcal{O}}_U^r$. Finite-rank locally free sheaves correspond to vector bundles in the standard geometric settings.

A free sheaf of finite rank on a scheme is a sheaf of modules globally isomorphic to ${\mathcal{O}}_X^{\oplus r}$. A locally free sheaf only requires such an isomorphism on an open cover; its local frames need not glue to one global frame.

A rank-one subsheaf of a vector bundle on a smooth algebraic curve is saturated when the quotient is torsion-free. Intersecting a one-dimensional subspace of the generic fibre with each stalk constructs a saturated subsheaf. The local ring of a smooth algebraic curve is a discrete valuation ring, so the subsheaf and its torsion-free quotient are locally free. Saturating a nonzero map from a line bundle enlarges that line bundle by an effective divisor, recording the map's zeros.

For a rank-two bundle $E$, the sufficiently positive twist $E(m{p}_0)$ is globally generated. The nowhere-vanishing section of a globally generated bundle on a curve gives a subbundle ${\mathcal{O}}_C\subset E(m{p}_0)$. Twisting back produces a genuine line subbundle of degree $-m$, with locally free quotient. Thus line-subbundle degrees are bounded above but not below. Merely inserting a negative divisor on an algebraic curve into a fixed subline would allow a torsion quotient and would not establish this stronger conclusion.

Choose a rational basis of the dual of a vector bundle and an effective divisor $D$ bounding all its poles. Evaluation gives a generically injective map $E\to {\mathcal{O}}_C(D{)}^{\oplus r}$. A line subbundle maps nontrivially to at least one summand, so ${\mathcal{O}}_C(D)\otimes L^{-1}$ has a nonzero regular section. Consequently $\mathrm{deg}L\le \mathrm{deg}D$. This bound depends on $E$ and the chosen rational frame, and does not require $E$ to be stable.

In a short exact sequence of sheaves $0\to \mathcal{F}\to \mathcal{G}\to \mathcal{H}\to 0$, if $\mathcal{H}$ is locally free of finite rank, the sequence splits near every point. Trivialize $\mathcal{H}$, lift its finitely many basis sections locally, and shrink to one neighbourhood on which all lifts exist. The lifts define a splitting. If $\mathcal{F}$ is also locally free, then so is $\mathcal{G}$. Conversely locally free kernel and middle term need not give a locally free quotient: multiplication by $t$ on ${\mathcal{O}}_{\mathrm{Spec}k[t]}$ has quotient supported at the origin.

A line bundle on a scheme is a locally free sheaf of rank one. Its global sections can define a morphism to projective space when they have no common zero.

On a smooth projective curve, a nonzero rational section represents a line bundle by its divisor on an algebraic curve. The degree is the sum of its vanishing orders minus its pole orders, including residue-field degrees over a nonclosed field. Principal divisors on an algebraic curve have degree zero, so this number is independent of the section. A nonzero regular section gives an effective divisor and therefore a nonnegative degree. Degrees add under tensor products and change sign under duality.

A half-density transforms by the square root of the absolute Jacobian under a coordinate change. The product of a half-density with its complex conjugate is an integrable density, so its norm is coordinate-independent without choosing a separate volume form. Half-densities provide an intrinsic configuration-space Hilbert space for a real cotangent polarization.

The orientation line bundle of a smooth $n$-manifold is the real line bundle ${\bigwedge }^{n}TM$. Its transition functions have the signs of the tangent-coordinate transition determinants. A nonzero continuous section specifies an orientation of the manifold. Its orientation double cover records the two possible local orientations.

The total space of the orientation line bundle is orientable even when its base is not: an orientation reversal in a base coordinate transition is matched by a reversal in the line coordinate, so their product has positive sign. Fiberwise multiplication by $1-t$ gives a deformation retraction onto the zero section. For the real projective plane this total space is $(S^2\times \mathbb{R})/((v,t)\sim (-v,-t))$, an oriented noncompact three-manifold homotopy equivalent to the plane.

On $S^2=SU(2)/U(1)$ a degree-$q$ equivariant line bundle corresponds to isotropy character $\mathrm{diag}(e^{it},e^{-it})\mapsto e^{-iqt}$. The Peter-Weyl theorem computes its section multiplicities by the occurrence of weight $-q$ in the SU(2) representation ${\mathrm{Sym}}^n{\mathbb{C}}^2$. Its weights $n,n-2,\dots ,-n$ give the displayed multiplicity-one decomposition. The spinor lines have degrees $-1,+1$ and thus contain all odd $n$, once each.

A line bundle trivial on each open set is determined by unit-valued transition functions on pairwise intersections. Their cocycle identities make the local trivial bundles glue; changing frames changes the cocycle by a coboundary. The tensor product of sheaves multiplies the transition functions, giving the displayed group isomorphism with Čech cohomology.

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 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.

The tensor product of two globally generated line bundles is globally generated. At a point choose generator germs arising from a global section of each factor. Their tensor is the germ of a global tensor-product section and generates the rank-one tensor-product stalk. Iteration gives the same property for every positive tensor power, on an arbitrary scheme.

On an integral scheme with generic point $\eta$ and function field $K$, a rational section of a line bundle $\mathcal{L}$ is an element of the one-dimensional $K$-space ${\mathcal{L}}_{\eta }$. A nonzero rational section writes as $a_i{e}_i$ in local trivializations, with $a_i\in K^{\ast }$. Ratios of these coefficients are regular units, and therefore define a Cartier divisor.

The twisting sheaf $\mathcal{O}(d)$ on $\mathrm{Proj}k[x_0,\dots ,x_n]$ is associated with the shifted graded module $k[x_0,\dots ,x_n](d)$. For $d\ge 0$, its global sections are the homogeneous polynomials of degree $d$.

On projective space the twisting sheaf $\mathcal{O}(d)$ is the $d$th tensor power of $\mathcal{O}(1)$ for $d\ge 0$, and the corresponding dual power for $d<0$. Twisting another sheaf means tensoring it with this line bundle.

The local frame $X_i^m$ on $X_i\ne 0$ identifies a section with a homogeneous rational function of degree $m$ on the punctured affine cone. Homogenizing a local rational expression and cancelling common factors writes every nonzero section as a quotient of coprime homogeneous polynomials $F/G$ with $\mathrm{deg}F-\mathrm{deg}G=m$. Conversely, such a rational function regular on the cone gives local coefficients $r/X_i^m$, which glue with the twisting sheaf's transition functions of a vector bundle.