Algebraic geometry

A presheaf of sets on a topological space

X

is a contravariant functor from the inclusion-ordered category

O (X)

of open subsets to

Set

. Its maps

F (U) \to F (V)

for

V \subseteq U

are restriction maps.

Constant presheaf of sets ( $K_{T} (U) = T$ )

Every restriction is the identity. Its stalk at each point is

T

, and its étale space of a presheaf is

X \times T

with

T

discrete. It is generally not a sheaf; in particular its value on the empty open set is

T

. Its sheafification is the constant sheaf of sets.

Morphism of presheaves ( $η : F \to G$ )

A natural transformation between presheaves assigns maps

F (U) \to G (U)

commuting with every restriction. For presheaves of abelian groups or modules, those maps preserve the specified algebraic structure.

Sheaf of sets

A sheaf of sets is a presheaf in which compatible sections on any open cover glue to one unique section on the union.

Skyscraper sheaf of sets ( $i_{} S (V) = S if x \in V, i_{} S (V) = 1 otherwise$ )

For inclusion of a point

i : {x} \to X

, this is the direct-image sheaf of a set. It is right adjoint to the stalk functor. It uses the singleton terminal set on opens omitting

x

, rather than the zero group used for abelian-group skyscraper sheaves.

Étale space of a presheaf ( $Et (P) = ∐_{x \in X} P_{x}$ )

The space of germs of a presheaf has basic opens given by the germs of one section over an open set. Projection to its base point is a local homeomorphism. Continuous sections of this projection give sheafification. For the constant presheaf of sets, the basic opens are

U \times {t}

, giving the ordinary product with a discrete fibre.

Constant sheaf of sets ( $Δ T (U) = LC (U, T)$ )

The sections are locally constant functions into a discrete set

T

, with ordinary restriction. It is the sheafification of the constant presheaf of sets. Its value on an empty open set is a singleton, even if

T

is empty. For a topological space, this construction is left adjoint to the global sections functor: a map from it to a sheaf is a family of global sections indexed by

T

Morphism of sheaves ( $ϕ : F \to G$ )

A morphism of sheaves assigns a map

F (U) \to G (U)

on every open set, commuting with all restriction maps. For sheaves of abelian groups, sheaves of rings or sheaves of modules, the maps must also preserve the relevant algebraic structure. It induces maps on every stalk, and equality of sheaf morphisms can be checked there.

f-morphism of sheaves ( $ϕ (U) : G (U) \to F (f^{- 1} U)$ )

For a continuous map

f : X \to Y

, an f-morphism from a sheaf on

Y

to a sheaf on

X

is a family of the displayed maps commuting with restrictions. It is equivalently a morphism of sheaves

G \to f_{*} F

. For each

x \in X

it induces a map

G_{f (x)} \to F_{x}

on stalks, by pulling back the neighbourhood representing a germ.

Surjective morphism of sheaves

A morphism of sheaves of modules is surjective when the induced map on every stalk is onto. Equivalently every target section can locally be lifted. It need not be onto on global sections; the long exact sequence in sheaf cohomology describes that obstruction.

Kernel sheaf ( $ker ϕ$ )

For a morphism of sheaves of abelian groups or modules, the kernel sheaf has sections

ker (F (U) \to G (U))

. These already satisfy the sheaf gluing axiom. Its stalk at

x

is the kernel of

F_{x} \to G_{x}

, since filtered colimits of modules preserve exactness. A sheaf morphism is injective exactly when its kernel sheaf is zero.

Sheaf of abelian groups

A sheaf of abelian groups is a sheaf of sets whose sections on each open set form an abelian group and whose restriction maps are group homomorphisms. Its stalk of a sheaf is the corresponding group of germs. Kernels are computed on sections, whereas cokernels are obtained by sheafification of the sectionwise cokernel. An exact sequence can therefore be checked on stalks.

Direct sum of sheaves ( $⨁_{i} F_{i}$ )

The direct sum of sheaves of abelian groups is the sheafification of the presheaf

U \mapsto ⨁_{i} 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.

Injective sheaf

A sheaf of abelian groups is injective if every morphism to it from a subsheaf extends to the containing sheaf. Every such sheaf embeds in an injective sheaf. Injective sheaves are flasque: the inclusion of extension by zero constant-integer sheaves associated to

U \subseteq V

turns the extension property into surjectivity of restriction from

V

U

Exact sequence of sheaves

An exact sequence of sheaves of abelian groups is a sequence of sheaf homomorphisms for which the image of each map equals the kernel of the next. Exactness can be checked on every stalk of a sheaf. Exactness of the associated sequences of sections on every open set is a stronger condition: a surjective sheaf homomorphism need only admit lifts locally.

Short exact sequence of sheaves

A sequence

0 \to F^{'} \to F \to F^{''} \to 0

of sheaves is exact exactly when the induced sequence on every stalk is exact.

Constant sheaf ( $\underline{A}$ )

For an abelian group

A

, the constant sheaf has sections over an open set given by its locally constant

A

-valued functions. It is the sheafification of the constant presheaf, and its sections on a disconnected open set need not have a single common value.

Sheaf gluing axiom

The sheaf gluing axiom says that sections

s_{i}

on an open cover

(U_{i})

that agree on every overlap

U_{i} \cap U_{j}

glue to a unique section on

⋃_{i} U_{i}

Ringed space

A ringed space

(X, O_{X})

is a topological space

X

equipped with a sheaf of rings

O_{X}

. A morphism consists of a continuous map

f : X \to Y

and a compatible morphism

f^{- 1} O_{Y} \to O_{X}

Morphism of ringed spaces

A morphism of ringed spaces

f : (X, O_{X}) \to (Y, O_{Y})

consists of a continuous map

f : X \to Y

and a morphism of sheaves of rings

f^{- 1} O_{Y} \to O_{X}

Sheaf of rings

A sheaf of rings assigns a ring

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.

Sheaf of modules

For a ringed space

(X, O_{X})

, a sheaf of

O_{X}

-modules is a sheaf

F

such that every

F (U)

is an

O_{X} (U)

-module and restriction maps preserve scalar multiplication.

Coherent analytic sheaf ( $F a coherent O_{X} -module$ )

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

O^{r} \to 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.

Dual of a sheaf ( $F^{\lor} = H o m_{O_{X}} (F, O_{X})$ )

The dual is the sheaf of modules whose sections on

U

are module-sheaf morphisms

F ∣_{U} \to 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.

Injective sheaf of modules

On a ringed space

(X, O_{X})

, an injective module sheaf is an injective object in the category of sheaves of modules over

O_{X}

. Its defining extension property concerns module-sheaf morphisms; it is distinct from injectivity solely in the category of sheaves of abelian groups.

Injective module sheaves are flasque

For opens

U \subseteq V \subseteq X

, there is a monomorphism between the extensions by zero of

O_{U}

and

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.

Stalk of a sheaf ( $F_{x}$ )

The stalk

F_{x}

is the direct limit of

F (U)

over neighborhoods

U

x

. A morphism of sheaves is an isomorphism or forms an exact sequence exactly when it does so on every stalk.

Stalk functor for presheaves of sets ( $F \mapsto F_{x}$ )

For presheaves of sets,

F_{x} = lim_{U ∋ 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.

Joint stalk functor on sheaves of sets ( $F \mapsto (F_{x})_{x \in X}$ )

The joint stalk functor

Sh (X) \to 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.

Skyscraper sheaf

For a point inclusion

i : {x} ↪ X

and an abelian group

A

, the skyscraper sheaf

i_{*} A

has sections

A

on opens containing

x

and zero on other opens. Its restriction maps are identities or maps to zero. If

A \neq = 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.

Skyscraper sheaf cohomology ( $H^{0} (X, i_{} A) = A, H^{i > 0} (X, i_{} A) = 0$ )

For any point inclusion

i : {x} ↪ X

, the skyscraper sheaf

i_{*} 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.

Germ of a sheaf section ( $s_{x} \in F_{x}$ )

The germ

s_{x}

of a section

s \in F (U)

is its image in the stalk of a sheaf

F_{x}

. It is zero exactly when

s

restricts to zero on some neighbourhood of

x

Support of a sheaf section ( $Supp s$ )

The support of a sheaf section is the closed set

Supp s = {x : s_{x} \neq = 0} .

Its complement is open because a zero germ of a sheaf section is represented by a section that vanishes on a neighbourhood.

Sheafification

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

Universal property of sheafification ( $Hom (F^{+}, H) ≅ Hom_{pre} (F, H)$ )

Every presheaf morphism from

F

to a sheaf

H

factors uniquely through the canonical sheafification map. Local representatives give local images in

H

, and the sheaf gluing axiom makes their union unique.

Sheafification by locally representable germs ( $F^{+} (U) \subseteq \prod_{x \in U} F_{x}$ )

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.

Extension by zero ( $j_{!} F$ )

For an open inclusion

j : U ↪ X

, extension by zero is the sheaf

j_{!} F

whose stalk equals

F_{x}

U

and zero outside

U

. Its sections are local sections on

U

whose support is closed in the ambient open set.

Quasi-coherent sheaf

A sheaf of modules on a scheme is quasi-coherent when every affine chart

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.

Affine module-sheaf equivalence ( $A -Mod ≃ QCoh (X)$ )

On an affine variety or affine scheme with coordinate ring

A

, the global section functor and

M \mapsto 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.

Torsion-free sheaf

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.

Vanishing of quasi-coherent cohomology on an affine scheme ( $H^{i > 0} (Spec A, M) = 0$ )

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.

Coherent sheaf

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.

Coherent ideal sheaf ( $I \subseteq O_{X}$ )

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.

Finite twisting resolution of a coherent sheaf on projective space ( $0 \to ⨁ O (a_{r j}) \to \dots \to ⨁ O (a_{0 j}) \to F \to 0$ )

A coherent sheaf on

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.

Support dimension of a coherent sheaf ( $dim Supp F$ )

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.

Associated point of a coherent sheaf

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.

Direct image of a coherent sheaf under a closed immersion

i : Z ↪ X

is a closed immersion of Noetherian schemes and

F

is a coherent sheaf on

Z

, then

i_{*} F

is coherent on

X

. Affine-locally this regards a finitely generated

A / I

-module as a finitely generated

A

-module.

Tensor product of sheaves ( $F \otimes_{O_{X}} G$ )

The tensor product of two sheaves of

O_{X}

-modules is the sheafification of

U \mapsto F (U) \otimes_{O_{X} (U)} G (U)

. Its stalk at

x

F_{x} \otimes_{O_{X, x}} G_{x}

Direct image sheaf ( $f_{*} F$ )

For a continuous map

f : X \to Y

and a sheaf

F

X

, the direct image is defined by

(f_{*} F) (V) = F (f^{- 1} V)

for every open

V \subseteq Y

Coherence reflected by finite direct image ( $F coherent ⟺ f_{*} F coherent$ )

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

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

Direct image from an open restriction ( $_{U} F = j_{*} (F ∣_{U})$ )

For an open inclusion

j : U ↪ X

, this sheaf has sections

F (V \cap U)

on an open set

V

. Restriction defines

F \to_{U} F

. Its stalks outside

U

may be nonzero, so it differs from extension by zero.

Direct-image tensor comparison ( $f_{} F \otimes_{O_{Y}} f_{} G \to f_{*} (F \otimes_{O_{X}} G)$ )

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.

Sheaf cohomology under a closed inclusion ( $H^{i} (Y, i_{*} F) ≅ H^{i} (X, F)$ )

For a closed inclusion

i : X ↪ 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.

Direct image of a quasi-coherent sheaf under an affine morphism

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

M

into the module sheaf associated with the same underlying module over the target ring.

Acyclic direct image from an affine chart of the projective line

Let

j : A_{k}^{1} ↪ P_{k}^{1}

be a standard affine chart and

G = 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} (P_{k}^{1}, j_{*} G) = 0

for

i > 0

. This does not assert that the direct image itself is flasque.

Quasi-coherence of direct image under a quasi-compact quasi-separated morphism

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.

Inverse image sheaf ( $f^{- 1} E$ )

The inverse image sheaf

f^{- 1} E

is the sheaf associated with the presheaf whose sections near

U \subseteq X

are obtained as a colimit of

E (V)

over open sets

V \supseteq f (U)

Inverse-image direct-image adjunction ( $f^{- 1} ⊣ f_{*}$ )

For sheaves of abelian groups and a continuous map

f : X \to Y

, there is a natural bijection

Hom_{X} (f^{- 1} G, F) ≅ Hom_{Y} (G, f_{*} 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.

Universal property of an inverse image sheaf ( $G \to f_{*} f^{- 1} G$ )

Every f-morphism of sheaves from

G

F

factors uniquely through the canonical f-morphism from

G

to its inverse image sheaf. A local inverse-image section is represented by sections of

G

on image neighbourhoods; their prescribed images glue in

F

Pullback of a sheaf of modules ( $f^{*} E$ )

For a morphism of ringed spaces, the pullback of an

O_{Y}

-module is

f^{*} E = O_{X} \otimes_{f^{- 1} O_{Y}} f^{- 1} E .

Restriction of a module sheaf to a closed subvariety ( $M ∣_{Spec (A / I)} = M / I M$ )

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 / I M

. This distinction matters because restricting only the underlying topological sheaf would retain the ambient ring action.

Pullback-direct-image adjunction unit ( $η_{H} : H \to f_{} f^{} H$ )

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

1 \otimes h

after inverse image. On the structure sheaf it becomes the given ring homomorphism

O_{Y} \to f_{*} O_{X}

Projection formula for sheaves

For a morphism of ringed spaces, there is a natural morphism

f_{*} F \otimes_{O_{Y}} E ⟶ f_{*} (F \otimes_{O_{X}} f^{*} E) .

It is an isomorphism when

E

is a locally free sheaf of finite rank, because the claim is local and then reduces to distributivity over a finite direct sum.

Locally free sheaf

A sheaf of

O_{X}

-modules is locally free of finite rank when every point has an open neighborhood

U

on which it is isomorphic to

O_{U}^{\oplus r}

for some finite

r

. If

X

is connected, the rank is constant.

Local splitting when a quotient sheaf is locally free

In a short exact sequence of sheaves

0 \to F \to G \to H \to 0

, if

H

is locally free of finite rank, the sequence splits near every point. Trivialize

H

, lift its finitely many basis sections locally, and shrink to one neighbourhood on which all lifts exist. The lifts define a splitting. If

F

is also locally free, then so is

G

. Conversely locally free kernel and middle term need not give a locally free quotient: multiplication by

t

O_{Spec k [t]}

has quotient supported at the origin.

Line bundle

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.

Line bundles trivialized by an open cover ( $Pic (X)_{U} ≅ \overset{ˇ}{H}^{1} (U, O_{X}^{*})$ )

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.

Globally generated line bundle

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.

Globally generated line bundle without finite generators ( $X = ∐_{m \geq 1} P^{m}, L ∣_{P^{m}} = O (1)$ )

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

P^{m}

; for

m \geq N

, these have a common zero. This shows why finite global generation on a quasi-compact scheme requires its finiteness hypothesis.

Finite global generation on a quasi-compact scheme

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.

Tensor product of globally generated line bundles

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.

Rational section of a line bundle

On an integral scheme with generic point

η

and function field

K

, a rational section of a line bundle

L

is an element of the one-dimensional

K

-space

L_{η}

. A nonzero rational section writes as

a_{i} e_{i}

in local trivializations, with

a_{i} \in K^{*}

. Ratios of these coefficients are regular units, and therefore define a Cartier divisor.

Twisting sheaf on projective space ( $O (d)$ )

The twisting sheaf

O (d)

Proj k [x_{0}, \dots, x_{n}]

is associated with the shifted graded module

k [x_{0}, \dots, x_{n}] (d)

. For

d \geq 0

, its global sections are the homogeneous polynomials of degree

d

Laurent-monomial description of top cohomology on projective space ( $H^{n} (P^{n}, O (m))$ )

For

n \geq 1

, the top Čech cohomology of the standard affine cover of projective space is the degree-

m

quotient of the Laurent ring by the sum of rings in which at least one variable has not been inverted. Its basis consists of Laurent monomials

X_{0}^{a_{0}} \dots X_{n}^{a_{n}}

with every

a_{i} < 0

and

\sum_{i} a_{i} = m

. Hence it vanishes for

m \geq - n

and has dimension

(n - m - 1)

for

m \leq - n - 1

Canonical bundle of projective space ( $ω_{P^{n}} ≅ O (- n - 1)$ )

Over any field, the canonical bundle of projective space is

O (- n - 1)

. Indeed, dualizing the Euler sequence gives

0 \to Ω_{P^{n}}^{1} \to O (- 1)^{\oplus (n + 1)} \to O \to 0

; taking determinants gives the result. Hence Serre duality identifies

H^{i} (P^{n}, O (m))

with the dual of

H^{n - i} (P^{n}, O (- m - n - 1))

Canonical-form transition on projective space ( $ω_{i} = (- 1)^{i} (X_{i} / X_{0})^{- n - 1} ω_{0}$ )

On the standard charts of projective space, wedge the coordinate differentials in increasing order. The change

X_{0} / X_{i} = (X_{i} / X_{0})^{- 1}

and

X_{j} / X_{i} = (X_{j} / X_{0}) / (X_{i} / X_{0})

gives the displayed Jacobian determinant. Multiplying the

i

th local generator by

(- 1)^{i}

removes this sign and yields the transition functions of the twisting sheaf on projective space with degree

- n - 1

Hyperplane exact sequence for twisting sheaves ( $0 \to O (m - 1) \to O (m) \to i_{*} O_{H} (m) \to 0$ )

For a hyperplane

H \subset P^{n}

with inclusion

i

, multiplication by its defining linear homogeneous polynomial gives the short exact sequence of sheaves

0 \to O_{P^{n}} (m - 1) \to O_{P^{n}} (m) \to i_{*} O_{H} (m) \to 0.

The long exact sequence in sheaf cohomology relates twists in consecutive degrees and consecutive dimensions. Surjectivity of restriction on global homogeneous polynomials, together with eventual positive-twist vanishing, proves vanishing of intermediate cohomology by descending induction. The top-degree dimensions satisfy

h_{n} (m - 1) = h_{n} (m) + h_{n - 1} (m)

for

n \geq 2

, where

h_{r} (m) = dim H^{r} (P^{r}, O (m))

Top-twist vanishing by hyperplane induction ( $H^{n} (P^{n}, O (m)) = 0 (m \geq - n, n > 0)$ )

The hyperplane exact sequence for twisting sheaves identifies consecutive top-degree groups whenever the hyperplane's preceding-degree group vanishes. Induction on dimension, starting with the surjective restriction of constants to a point on

P^{1}

, propagates the degree-zero top vanishing down to

m = - n

. Its top-degree surjections also propagate vanishing to every positive twist. This proof does not apply the negative-twist global-section formula to

P^{0}

Picard group ( $Pic (X)$ )

The Picard group is the group of isomorphism classes of line bundles under tensor product.

Holomorphic Picard group ( $Pic_{hol} (X) = H^{1} (X, O_{X}^{*})$ )

The holomorphic Picard group consists of isomorphism classes of holomorphic line bundles, with tensor product as its group law. The trivial bundle is the identity and dualization gives inverses. Transition functions identify it with

H^{1} (X, O_{X}^{*})

. On the complex projective line every class is

O (n)

, so degree gives an isomorphism with

Z

Picard injection for holomorphic projective bundles ( $(M, n) \mapsto p^{*} M \otimes L^{\otimes n}$ )

For a rank-two holomorphic vector bundle on a nonempty complex manifold, this map from

Pic_{hol} (X) \times Z

Pic_{hol} (P (E))

is injective. Fibre degree detects

n

. If

p^{*} M

is trivial, a nonvanishing holomorphic section is constant along every compact projective fibre in a local bundle chart, so it descends to a nonvanishing section of

M

. The proof requires no global section of

p

Picard-group localization on a smooth variety ( $⨁_{j} Z [D_{j}] \to Pic (X) \to Pic (U) \to 0$ )

For a smooth integral variety, every Weil divisor is a Cartier divisor, so the Picard group equals the divisor class group. If

U

is obtained by removing irreducible Weil divisors

D_{j}

, the localization sequence for the divisor class group gives the displayed exact sequence. A rational trivialization of a line bundle on

U

has divisor supported on the removed divisors, describing the kernel concretely.

Cartier-divisor description of the Picard group ( $Pic (X) ≅ Γ (X, K_{X}^{} / O_{X}^{}) / k (X)^{*}$ )

On an irreducible variety, a Cartier divisor is a section of

K_{X}^{*} / O_{X}^{*}

: its local rational equations differ by regular units. Their ratios define an invertible sheaf, and a global rational equation gives the trivial class. Conversely a rational trivialization of an invertible sheaf gives local equations. The rational-function sheaf is flasque, so its long exact sequence in sheaf cohomology also identifies the quotient with

H^{1} (X, O_{X}^{*})

Néron-Severi group ( $NS (X)$ )

The Néron-Severi group is the quotient

Pic (X) / Pic^{0} (X)

of the Picard group by line bundles algebraically equivalent to zero.

Very ample line bundle

A line bundle is very ample when its global sections define a closed embedding into projective space.

Ample line bundle

A line bundle

L

is ample when some positive tensor power

L^{\otimes m}

is very ample.

Ampleness on reduced components

On a projective scheme, a line bundle is ample exactly when its restriction to every reduced irreducible component is ample. The disjoint union of those components maps finitely and surjectively to the scheme, so finite-surjective descent of the finite pullback of an ample line bundle proves this. The same argument applies to restrictions to any closed subscheme. Thus nilpotent structure does not affect ampleness.

Finite pullback of an ample line bundle

For a finite morphism

f : X \to Y

, the pullback of an ample line bundle is ample. If

f

is also surjective, ampleness is equivalent in both directions. These results hold for proper schemes over a field, including nonreduced schemes. Cohomological ampleness and the exact finite pushforward prove the forward direction; coherent-sheaf devissage gives finite-surjective descent. In particular, a finite map to projective space makes the pulled-back hyperplane bundle ample.

High ample twist is very ample

A

is an ample line bundle on a projective variety and

L

is any line bundle, then

A^{\otimes m} \otimes L

is very ample for all sufficiently large integers

m

. Taking

L

to be trivial also makes

A^{\otimes m}

very ample.

Kodaira map

A basepoint-free vector space of global sections

V \subseteq H^{0} (X, L)

defines the Kodaira map

X \to P (V^{*})

by evaluating the sections at each point.

Global section functor ( $Γ (X, -)$ )

The global section functor sends a sheaf

F

X

Γ (X, F)

. It is left exact, and its right derived functors are sheaf cohomology.

Localization of global sections on a principal open ( $Γ (X, O_{X})_{f} ≅ Γ (X_{f}, O_{X})$ )

For a quasi-compact separated scheme and a global regular function

f

, global sections on

X_{f}

are the localization of global sections by

f

. Choose a finite affine open cover. Its intersections are affine, so the sheaf gluing axiom describes global sections as the kernel of the difference map between two finite products of affine coordinate rings. Exactness of localization preserves this kernel, and localization commutes with those finite products. Localizing the same cover gives the claimed isomorphism. This applies before knowing whether

X

itself is affine.

Global section ( $s \in Γ (X, F)$ )

A global section of a sheaf of modules

F

is an element of

Γ (X, F)

, meaning a section defined over the whole space

X

Section functor with support ( $Γ_{Z} (X, -)$ )

For a closed subset

Z \subseteq X

, the section functor with support is

Γ_{Z} (X, F) = {s \in Γ (X, F) : Supp s \subseteq Z} .