Ringed space

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.

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.

Sheafification

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

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.

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.

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.

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

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)

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 .

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.

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.

Picard group ( $Pic (X)$ )

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

Very ample line bundle

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

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.

Sheaf cohomology ( $H^{p} (X, F)$ )

Sheaf cohomology consists of the right derived functors of global sections. The zeroth group is

H^{0} (X, F) = Γ (X, F)

, and higher groups measure obstructions to gluing local sections.

Flasque sheaf

A sheaf is flasque when every restriction map is surjective. Flasque sheaves are acyclic for global sections, so

H^{i} (X, F) = 0

for

i > 0

Čech cohomology ( $\overset{ˇ}{H}^{p} (U, F)$ )

For an open cover

U = (U_{i})

, the Čech cochain group is

\overset{ˇ}{C}^{p} (U, F) = \prod_{i_{0} < \dots < i_{p}} F (U_{i_{0}} \cap \dots \cap U_{i_{p}}),

with the alternating sum of restrictions as differential. Its cohomology is the Čech cohomology of

F

with respect to

U

Čech cochain complex ( $\overset{ˇ}{C}^{∙} (U, F)$ )

The Čech cochain complex places sections on

(p + 1)

-fold intersections in degree

p

and uses the alternating sum of restriction maps as its differential.

Čech coboundary

A Čech coboundary is the image of a cochain in the preceding degree. Multiplicatively, a zero-cochain

(h_{i})

changes a one-cocycle by

g_{ij} \mapsto h_{i}^{- 1} g_{ij} h_{j}

Acyclic cover theorem

If every nonempty finite intersection of members of an open cover has vanishing higher sheaf cohomology for

F

, then the cover's Čech cohomology computes

H^{p} (X, F)

. An affine open cover of a separated scheme satisfies this condition for a quasi-coherent sheaf because its finite intersections are affine.

Mayer-Vietoris sequence for sheaf cohomology

For

X = U \cup V

, the short exact sequence of sheaves obtained by restricting to

U

V

, and

U \cap V

induces a long exact sequence

0 \to H^{0} (X, F) \to H^{0} (U, F) \oplus H^{0} (V, F) \to H^{0} (U \cap V, F) \to H^{1} (X, F) \to \dots .

Locally ringed space

A locally ringed space is a ringed space whose stalk

O_{X, x}

is a local ring at every point. Morphisms of locally ringed spaces induce local homomorphisms on stalks.

Scheme

A scheme is a locally ringed space covered by open subsets isomorphic to spectra of commutative rings. It retains both the points defined by prime ideals and the local algebra of functions around them.

Spectrum of a commutative ring ( $Spec A$ )

The spectrum

Spec A

of a commutative ring

A

is the set of its prime ideals, with closed sets

V (I) = {p : I \subseteq p}

and a structure sheaf whose sections locally look like fractions.

Affine scheme

An affine scheme is a scheme isomorphic to

Spec A

for some commutative ring

A

. Homomorphisms

A \to B

correspond contravariantly to morphisms

Spec B \to Spec A

Principal open subscheme ( $D (f)$ )

For

f \in A

, the principal open subset

D (f) = {p : f \in / p}

Spec A

is the affine scheme

Spec A_{f}

Affine plane ( $A_{k}^{2}$ )

The affine plane over a field

k

is the affine scheme

Spec k [x, y]

Punctured affine plane ( $A_{k}^{2} ∖ {0}$ )

The punctured affine plane is obtained by deleting the closed point

(x, y)

from

Spec k [x, y]

. Its regular functions still form

k [x, y]

, while

H^{1} (A_{k}^{2} ∖ {0}, O) ≅ k [x^{\pm 1}, y^{\pm 1}] / (k [x^{\pm 1}, y] + k [x, y^{\pm 1}]),

which has basis represented by

x^{- a} y^{- b}

for

a, b \geq 1

Noetherian scheme

A Noetherian scheme has a finite cover by affine schemes

Spec A_{i}

with each

A_{i}

a Noetherian ring. Equivalently, it is quasi-compact and locally Noetherian.

Morphism of schemes

A morphism of schemes is a morphism of locally ringed spaces. On affine schemes it is contravariantly equivalent to a homomorphism of their coordinate rings.

Module of Kähler differentials ( $Ω_{B / A}$ )

For a ring map

A \to B

, the module of Kähler differentials represents

A

-derivations:

Hom_{B} (Ω_{B / A}, M) ≅ Der_{A} (B, M)

Conormal exact sequence for Kähler differentials

For

A \to B \to C = B / I

, there is a right-exact sequence

I / I^{2} ⟶ Ω_{B / A} \otimes_{B} C ⟶ Ω_{C / A} ⟶ 0,

where

[i] \mapsto d i \otimes 1

and

d b \otimes c \mapsto c d \overset{ˉ}{b}

Separated morphism

A morphism

f : X \to Y

is separated when its diagonal

X \to X \times_{Y} X

is a closed immersion. This is the scheme-theoretic analogue of the Hausdorff property.

Valuative criterion for separatedness

For a finite-type morphism of Noetherian schemes, separatedness is equivalent to uniqueness in every lifting problem

Spec K \to X

over

Spec R \to Y

, where

R

is a valuation ring with fraction field

K

Proper morphism

A morphism is proper when it is separated, of finite type, and universally closed. Properness is stable under base change and composition.

Finite complex computing cohomology in a proper flat family

For a proper morphism

X \to Spec A

with

A

Noetherian and an

A

-flat coherent sheaf

F

, locally on the base there is a bounded complex of finite free modules

K^{∙}

such that

H^{p} (X, F \otimes_{A} M) ≅ H^{p} (K^{∙} \otimes_{A} M)

naturally for every

A

-module

M

Semicontinuity theorem for coherent cohomology

In a proper flat family with a coherent sheaf, the fiber dimension

s \mapsto dim_{κ (s)} H^{p} (X_{s}, F_{s})

is upper semicontinuous, and the fiberwise Euler characteristic is locally constant.

Valuative criterion for properness

For a finite-type morphism of Noetherian schemes, properness is equivalent to existence and uniqueness in every lifting problem from the generic point

Spec K

of a valuation ring

R

Spec R

Proj construction ( $Proj S$ )

For a graded ring

S

Proj S

consists of homogeneous prime ideals not containing the irrelevant ideal

S_{+}

. Its standard affine opens satisfy

D_{+} (f) ≅ Spec (S_{f})_{0}

Closed immersion

A morphism

i : Z \to X

is a closed immersion when it identifies

Z

homeomorphically with a closed subset of

X

and the morphism

O_{X} \to i_{*} O_{Z}

is surjective. Affine-locally it has the form

Spec (A / I) \to Spec A

Closed subscheme

A closed subscheme of

X

is a scheme

Z

together with a closed immersion

Z ↪ X

, usually identified with its image and its quotient structure sheaf.

Reduced induced subscheme

Every closed subset

Z \subseteq X

has a canonical reduced closed-subscheme structure defined affine-locally by

Spec (A / I)

when

Z = V (I) \subseteq Spec A

. It is the smallest closed subscheme with underlying set

Z

Scheme-theoretic image

The scheme-theoretic image of

f : X \to Y

is the smallest closed subscheme of

Y

through which

f

factors. For an affine morphism

Spec B \to Spec A

induced by

φ : A \to B

, it is

Spec (A / ker φ)

, whose underlying set is the closure of the set-theoretic image.

Affine plane with doubled origin