aarjaneiro/planetmath huggingface dataset:
dual isogeny

dual isogeny

Given an

isogeny

f

:

E

→

E

′

of

elliptic curves

of degree

n

, the

dual isogeny

is an isogeny

f

^

:

E

′

→

E

of the same degree such that

f

∘

f

^

=

[

n

]

. Here

[

n

]

denotes the multiplication-by-

n

isogeny

e

↦

n

⁢

e

which has degree

n

2

.

Often only the

existence

of a dual isogeny is needed, but the construction is explicit as

E

′

→

Div

0

⁡

(

E

′

)

→

f

*

Div

0

⁡

(

E

)

→

E

where

Div

0

is the group of

divisors

of degree 0.
To do this, we need maps

E

→

Div

0

⁡

(

E

)

given by

P

↦

P

-

O

where

O

is the neutral point of

E

and

Div

0

⁡

(

E

)

→

E

given by

∑

n

P

⁢

P

↦

∑

n

P

⁢

P

.

To see that

f

∘

f

^

=

[

n

]

, note that the original isogeny

f

can be written as a

composite

E

→

Div

0

⁡

(

E

)

→

f

*

Div

0

⁡

(

E

′

)

→

E

′

and that since

f

is finite of degree

n

,

f

*

⁢

f

*

is

multiplication

by

n

on

Div

0

⁡

(

E

′

)

.

Alternatively, we can use the smaller

Picard group

Pic

0

, a quotient of

Div

0

. The map

E

→

Div

0

⁡

(

E

)

descends to an

isomorphism

,

E

→

∼

Pic

0

⁡

(

E

)

. The dual isogeny is

E

′

→

∼

Pic

0

⁡

(

E

′

)

→

f

*

Pic

0

⁡

(

E

)

→

∼

E

Note that the

relation

f

∘

f

^

=

[

n

]

also implies the

conjugate

relation

f

^

∘

f

=

[

n

]

. Indeed, let

ϕ

=

f

^

∘

f

. Then

ϕ

∘

f

^

=

f

^

∘

[

n

]

=

[

n

]

∘

f

^

. But

f

^

is

surjective

, so we must have

ϕ

=

[

n

]

.

Title

dual isogeny

Canonical name

DualIsogeny

Date of creation

2013-03-22 12:52:58

Last modified on

2013-03-22 12:52:58

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Definition

Classification

msc 14-00

Related topic

ArithmeticOfEllipticCurves

Generated on Fri Feb 9 17:55:51 2018 by

LaTeXML
example of quasi-affine variety that is not affine

example of quasi-affine variety that is not affine

Let

k

be an algebraically closed field. Then the

affine plane

𝔸

2

is certainly affine. If we remove the point

(

0

,

0

)

, then we obtain a

quasi-affine variety

A

.

The ring of regular functions of

A

is the same as the ring of regular functions of

𝔸

2

. To see this, first observe that the two

varieties

are clearly birational, so they have the same function field. Clearly also any function

regular

on

𝔸

2

is regular on

A

. So let

f

be regular on

A

. Then it is a rational function on

𝔸

2

, and its poles (if any) have

codimension

one, which means they will have

support

on

A

. Thus it must have no poles, and therefore it is regular on

𝔸

2

.

We know that the

morphisms

A

→

𝔸

2

are in natural bijection with the morphisms from the

coordinate ring

of

𝔸

2

to the coordinate ring of

A

; so

isomorphisms

would have to correspond to

automorphisms

of

k

⁢

[

X

,

Y

]

, but this is just the set of

invertible linear transformations

of

X

and

Y

; none of these yield an isomorphism

A

→

𝔸

2

.

Alternatively, one can use Čech

cohomology

to show that

H

1

⁢

(

A

,

𝒪

A

)

is nonzero (in fact, it is infinite-dimensional), while every

affine variety

has zero higher

cohomology groups

.

For further information on this sort of subject, see Chapter I of Hartshorne’s (which lists this as exercise I.3.6). See the

bibliography for algebraic geometry

for this and other books.

Title

example of quasi-affine variety that is not affine

Canonical name

ExampleOfQuasiaffineVarietyThatIsNotAffine

Date of creation

2013-03-22 14:16:39

Last modified on

2013-03-22 14:16:39

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Example

Classification

msc 14-00

Generated on Fri Feb 9 17:55:53 2018 by

LaTeXML
line bundle

line bundle

In

algebraic geometry

, the term

line bundle

refers to a

locally free

coherent sheaf

of rank 1, also called an

invertible sheaf

. In manifold theory, it refers to a real or complex one dimensional

vector bundle

. These notions are

equivalent

on a

non-singular

complex algebraic variety

X

: given a one dimensional vector bundle, its sheaf of holomorphic

sections

is locally free and of rank 1. Similarly, given a locally free sheaf

ℱ

of rank one, the space

ℒ

=

∪

x

∈

X

ℱ

x

/

𝔪

x

⁢

ℱ

x

,

given the coarsest

topology

for which sections of

ℱ

define

continuous functions

in a vector bundle of complex dimension 1 over

X

, with the obvious map taking the stalk over a point to that point.

Title

line bundle

Canonical name

LineBundle

Date of creation

2013-03-22 13:31:04

Last modified on

2013-03-22 13:31:04

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Definition

Classification

msc 14-00

Generated on Fri Feb 9 17:55:59 2018 by

LaTeXML
nonsingular variety

nonsingular variety

A variety over an algebraically closed field

k

is

nonsingular

at a point

x

if the local ring

𝒪

x

is a regular local ring.
Equivalently, if around the point one has an open affine neighborhood
wherein the variety is cut out
by certain

polynomials

F

1

,

…

,

F

n

of

m

variables

x

1

,

…

,

x

m

,
then it is nonsingular at

x

if the

Jacobian

has maximal rank at that point.
Otherwise,

x

is a

singular point

.

A variety is

nonsingular

if it is nonsingular at each point.

Over the real or

complex numbers

, nonsingularity corresponds to “smoothness”:
at nonsingular points, varieties are locally real or

complex manifolds

(this is simply the

implicit function theorem

).
Singular points generally have “corners” or self

intersections

.
Typical examples are the curves

x

2

=

y

3

,
which has a cusp at

(

0

,

0

)

and is nonsingular everywhere else,
and

x

2

⁢

(

x

+

1

)

=

y

2

,
which has a self-intersection at

(

0

,

0

)

and is nonsingular everywhere else.

Title

nonsingular variety

Canonical name

NonsingularVariety

Date of creation

2013-03-22 12:03:47

Last modified on

2013-03-22 12:03:47

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 14-00

Synonym

non-singular variety

Defines

nonsingular

Defines

non-singular

Defines

singular point

Defines

nonsingular point

Defines

non-singular point

Generated on Fri Feb 9 17:56:00 2018 by

LaTeXML
Picard group

Picard group

The

Picard group

of a

variety

, scheme, or more generally

locally
ringed space

(

X

,

O

X

)

is the group of

locally free

O

X

modules of rank

1

with

tensor product

over

O

X

as the operation, usually denoted by

Pic

⁡

(

X

)

. Alternatively, the Picard group is the group of isomorphism classes of

invertible

sheaves on

X

, under tensor products.

It is not difficult to see that

Pic

⁡

(

X

)

is

isomorphic

to

H

1

⁢

(

X

,

O

X

*

)

, the
first

sheaf cohomology

group of the

multiplicative

sheaf

O

X

*

which consists of the
units of

O

X

.

Finally, let

CaCl

⁡

(

X

)

be the group of

Cartier divisors

on

X

modulo linear equivalence. If

X

is an integral scheme then the groups

CaCl

⁡

(

X

)

and

Pic

⁡

(

X

)

are isomorphic. Furthermote, if we let

Cl

⁡

(

X

)

be the

class group

of

Weil divisors

(

divisors

modulo

principal divisors

) and

X

is a

noetherian

,

integral

and

separated

locally

factorial

scheme, then there is a

natural isomorphism

Cl

⁡

(

X

)

≅

Pic

⁡

(

X

)

. Thus, the Picard group is sometimes called the

divisor class group

of

X

.

Title

Picard group

Canonical name

PicardGroup

Date of creation

2013-03-22 12:52:30

Last modified on

2013-03-22 12:52:30

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Definition

Classification

msc 14-00

Synonym

divisor class group

Generated on Fri Feb 9 17:56:02 2018 by

LaTeXML
projective variety

projective variety

Given a

homogeneous polynomial

F

of degree

d

in

n

+

1

variables

X

0

,

…

,

X

n

and a point

[

x

0

:

⋯

:

x

n

]

, we cannot evaluate

F

at that point, because it has

multiple

such

representations

, but since

F

⁢

(

λ

⁢

x

0

,

…

,

λ

⁢

x

n

)

=

λ

d

⁢

F

⁢

(

x

0

,

…

,

x

n

)

we can say whether any such representation (and hence all) vanish at that point.

A

projective variety

over an algebraically closed field

k

is a subset of some

projective space

ℙ

k

n

over

k

which can be described as the common vanishing locus of finitely many homogeneous polynomials with

coefficients

in

k

, and which is not the union of two such smaller loci. Also, a

quasi-projective variety

is an open subset of a projective variety.

Title

projective variety

Canonical

name

ProjectiveVariety

Date of creation

2013-03-22 12:03:58

Last modified on

2013-03-22 12:03:58

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Definition

Classification

msc 14-00

Related topic

AffineVariety

Related topic

Scheme

Related topic

AlgebraicGeometry

Related topic

Variety

Related topic

ChowsTheorem

Defines

quasi-projective variety

Generated on Fri Feb 9 17:56:07 2018 by

LaTeXML
quasi-finite morphism

quasi-finite morphism

A

morphism

f

:

X

→

Y

of schemes or

varieties

is

quasi-finite

if for each

y

∈

Y

, the fiber

f

-

1

⁢

(

y

)

is a

finite set

.

Title

quasi-finite morphism

Canonical name

QuasifiniteMorphism

Date of creation

2013-03-22 12:52:10

Last modified on

2013-03-22 12:52:10

Owner

nerdy2 (62)

Last modified by

nerdy2 (62)

Numerical id

4

Author

nerdy2 (62)

Entry type

Definition

Classification

msc 14-00

Classification

msc 14A10

Classification

msc 14A15

Synonym

quasi-finite

Generated on Fri Feb 9 17:56:08 2018 by

LaTeXML
variety

variety

Definition 1

Let

X

be a scheme over a field

k

. Then

X

is said to be an

abstract variety

over

k

if it is integral,

separated

, and of

finite type

over

k

. Usually we simply say

X

is a

variety

. If

X

is proper over

k

, it is said to be

complete

. If the

dimension

of

X

is one, then

X

is said to be a

curve

.

Some authors also require

k

to be

algebraically closed

, and some authors require curves to be

nonsingular

.

Calling

X

a variety would appear to conflict with the preexisting notion of an affine (

http://planetmath.org/AffineVariety

) or

projective variety

. However, it can be shown that if

k

is algebraically closed, then there is an

equivalence of categories

between affine abstract varieties over

k

and

affine varieties

over

k

, and another between projective abstract varieties over

k

and projective varieties over

k

.

This equivalence of categories identifies an abstract variety with the set of its

k

-points; this can be thought of as simply ignoring all the

generic points

. In the other

direction

, it identifies an affine variety with the

prime spectrum

of its

coordinate ring

: the variety in

𝔸

n

defined by the ideal

⟨

f

1

,

…

,

f

m

⟩

is identified with

Spec

⁡

k

⁢

[

X

1

,

…

,

X

n

]

/

⟨

f

1

,

…

,

f

m

⟩

.

A projective variety is identified as the gluing together of the affine varieties obtained by taking the

complements

of

hyperplanes

. To see this, suppose we have a projective variety in

ℙ

n

given by the

homogeneous ideal

⟨

f

1

,

…

,

f

m

⟩

. If we delete the hyperplane

X

i

=

0

, then we obtain an affine variety: let

T

j

=

X

j

/

X

i

; then the affine variety is the set of common zeros of

⟨

f

1

⁢

(

T

0

,

…

,

T

n

)

,

…

,

f

m

⁢

(

T

0

,

…

,

T

n

)

⟩

.

In this way, we can get

n

+

1

overlapping affine varieties that cover our original projective variety. Using the theory of schemes, we can glue these affine varieties together to get a scheme; the result will be projective.

For more on this, see Hartshorne’s book

Algebraic Geometry

; see the

bibliography for algebraic geometry

for more resources.

Title

variety

Canonical name

Variety

Date of creation

2013-03-22 14:16:43

Last modified on

2013-03-22 14:16:43

Owner

mps (409)

Last modified by

mps (409)

Numerical id

9

Author

mps (409)

Entry type

Definition

Classification

msc 14-00

Synonym

abstract variety

Related topic

Scheme

Related topic

AffineVariety

Related topic

ProjectiveVariety

Defines

complete

Defines

curve

Generated on Fri Feb 9 17:56:10 2018 by

LaTeXML
radical of an ideal

radical of an ideal

Let

R

be a

commutative ring

. For any ideal

I

of

R

, the

radical

of

I

, written

I

or

Rad

⁡

(

I

)

, is the set

{

a

∈

R

⁢

∣

a

n

∈

I

⁢

for some integer

⁢

n

>

⁢

0

}

The radical of an ideal

I

is always an ideal of

R

.

If

I

=

I

, then

I

is called a

radical ideal

.

Every

prime ideal

is a radical ideal. If

I

is a radical ideal, the

quotient ring

R

/

I

is a ring with no nonzero

nilpotent elements

.

More generally, the radical of an ideal in can be defined over an arbitrary ring. Let

I

be an ideal of a ring

R

, the radical of

I

is the set of

a

∈

R

such that every

m-system

containing

a

has a non-empty

intersection

with

I

:

I

:=

{

a

∈

R

∣

if

⁢

S

⁢

is an

m

-system, and

⁢

a

∈

S

,

then

⁢

S

∩

I

≠

∅

}

.

Under this definition, we see that

I

is again an ideal (two-sided) and it is a subset of

{

a

∈

R

⁢

∣

a

n

∈

I

⁢

for some integer

⁢

n

>

⁢

0

}

. Furthermore, if

R

is

commutative

, the two sets coincide. In other words, this definition of a radical of an ideal is indeed a “

generalization

” of the radical of an ideal in a commutative ring.

Title

radical of an ideal

Canonical name

RadicalOfAnIdeal

Date of creation

2013-03-22 12:35:54

Last modified on

2013-03-22 12:35:54

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

17

Author

CWoo (3771)

Entry type

Definition

Classification

msc 14A05

Classification

msc 16N40

Classification

msc 13-00

Related topic

PrimeRadical

Related topic

RadicalOfAnInteger

Related topic

JacobsonRadical

Related topic

HilbertsNullstellensatz

Related topic

AlgebraicSetsAndPolynomialIdeals

Defines

radical ideal

Defines

radical

Generated on Fri Feb 9 17:56:12 2018 by

LaTeXML
algebraic map

algebraic map

A map

f

:

X

→

Y

between

quasi-affine varieties

X

⊂

k

n

,

Y

⊂

k

m

over a field

k

is called

if there is a map

f

′

:

k

n

→

k

m

whose

component

functions are

polynomials

, such that

f

′

restricts to

f

on

X

.

Alternatively,

f

is

algebraic

if the pullback map

f

*

:

C

⁢

(

Y

)

→

C

⁢

(

X

)

maps the

coordinate ring

of

Y

,

k

⁢

[

Y

]

, to the coordinate ring of

X

,

k

⁢

[

X

]

.

Title

algebraic map

Canonical name

AlgebraicMap

Date of creation

2013-03-22 13:53:41

Last modified on

2013-03-22 13:53:41

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 14A10

Synonym

algebraic

Generated on Fri Feb 9 17:56:14 2018 by

LaTeXML
algebraic sets and polynomial ideals

algebraic sets and polynomial ideals

Suppose

k

is a field. Let

𝔸

k

n

denote affine

n

-space over

k

.

For

S

⊆

k

⁢

[

x

1

,

…

,

x

n

]

, define

V

⁢

(

S

)

, the

zero set

of

S

, by

V

⁢

(

S

)

=

{

(

a

1

,

…

,

a

n

)

∈

k

n

∣

f

⁢

(

a

1

,

…

,

a

n

)

=

0

⁢

for all

⁢

f

∈

S

}

We say that

Y

⊆

𝔸

k

n

is an (affine)

algebraic set

if there exists

T

⊆

k

⁢

[

x

1

,

…

,

x

n

]

such that

Y

=

V

⁢

(

T

)

. Taking these subsets of

𝔸

k

n

as a definition of the

closed sets

of a topology induces the

Zariski topology

over

𝔸

k

n

.

For

Y

⊆

𝔸

k

n

, define the

deal of

Y

in

k

⁢

[

x

1

,

…

,

x

n

]

by

I

⁢

(

Y

)

=

{

f

∈

k

⁢

[

x

1

,

…

,

x

n

]

∣

f

⁢

(

P

)

=

0

⁢

for all

⁢

P

∈

Y

}

.

It is easily shown that

I

⁢

(

Y

)

is an ideal of

k

⁢

[

x

1

,

…

,

x

n

]

.

Thus we have defined a

function

V

mapping

from subsets of

k

⁢

[

x

1

,

…

,

x

n

]

to algebraic sets in

𝔸

k

n

, and a function

I

mapping from subsets of

𝔸

n

to ideals of

k

⁢

[

x

1

,

…

,

x

n

]

.

We remark that the theory of algebraic sets presented herein is most cleanly stated over an algebraically closed field. For example, over such a field, the above have the following properties:

1.

S

1

⊆

S

2

⊆

k

⁢

[

x

1

,

…

,

x

n

]

implies

V

⁢

(

S

1

)

⊇

V

⁢

(

S

2

)

.

2.

Y

1

⊆

Y

2

⊆

𝔸

k

n

implies

I

⁢

(

Y

1

)

⊇

I

⁢

(

Y

2

)

.

3.

For any ideal

𝔞

⊂

k

⁢

[

x

1

,

…

,

x

n

]

,

I

⁢

(

V

⁢

(

𝔞

)

)

=

Rad

⁡

(

𝔞

)

.

4.

For any

Y

⊂

𝔸

k

n

,

V

⁢

(

I

⁢

(

Y

)

)

=

Y

¯

, the

closure

of

Y

in the Zariski topology.

From the above, we see that there is a

1-1 correspondence

between algebraic sets in

𝔸

k

n

and

radical ideals

of

k

⁢

[

x

1

,

…

,

x

n

]

. Furthermore, an algebraic set

Y

⊆

𝔸

k

n

is an

affine variety

if and only if

I

⁢

(

Y

)

is a

prime ideal

. As an example of how things can go wrong, the radical ideals

(

1

)

and

(

x

2

+

1

)

in

ℝ

⁢

[

x

]

define the same zero locus (the

empty set

) inside of

ℝ

, but are not the same ideal, and hence there is no such 1-1 correspondence.

Title

algebraic sets and polynomial ideals

Canonical name

AlgebraicSetsAndPolynomialIdeals

Date of creation

2013-03-22 13:05:40

Last modified on

2013-03-22 13:05:40

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

16

Author

mathcam (2727)

Entry type

Definition

Classification

msc 14A10

Synonym

vanishing set

Related topic

Ideal

Related topic

HilbertsNullstellensatz

Related topic

RadicalOfAnIdeal

Defines

zero set

Defines

algebraic set

Defines

ideal of an algebraic set

Defines

affine algebraic set

Generated on Fri Feb 9 17:56:16 2018 by

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alternative characterizations of Noetherian topological spaces

alternative characterizations of Noetherian topological spaces

Let

X

be a

topological space

. The following conditions are

equivalent

conditions for

X

to be a

Noetherian topological space

:

1.

X

satisfies

the

descending chain condition

(

http://planetmath.org/DescendingChainCondition

) for

closed subsets

.

2.

X

satisfies the

ascending chain condition

(

http://planetmath.org/AscendingChainCondition

) for

open subsets

.

3.

Every nonempty family of closed subsets has a

minimal element

.

4.

Every nonempty family of open subsets has a

maximal element

.

5.

Every subset of

X

is

compact

.

Title

alternative characterizations of Noetherian topological spaces

Canonical name

AlternativeCharacterizationsOfNoetherianTopologicalSpaces

Date of creation

2013-03-22 14:16:25

Last modified on

2013-03-22 14:16:25

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

10

Author

yark (2760)

Entry type

Theorem

Classification

msc 14A10

Generated on Fri Feb 9 17:56:18 2018 by

LaTeXML
Bézout’s theorem (Algebraic Geometry)

Bézout’s theorem (Algebraic Geometry)

The classic version of Bézout’s

theorem

states that two complex

projective curves

of degrees

m

and

n

which share no common component

intersect

in exactly

m

⁢

n

points if the points are counted with

multiplicity

.

The generalized version of Bézout’s theorem states that if

A

and

B

are

algebraic varieties

in

k

-dimensional

projective space

over an algebraically complete field and

A

∩

B

is a

variety

of

dimension

dim

⁢

(

A

)

+

dim

⁢

(

B

)

-

k

, then the degree of

A

∩

B

is the

product

of the degrees of

A

and

B

.

Title

Bézout’s theorem (

Algebraic Geometry

)

Canonical

name

BezoutsTheoremAlgebraicGeometry

Date of creation

2013-03-22 14:36:45

Last modified on

2013-03-22 14:36:45

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Algorithm

Classification

msc 14A10

Generated on Fri Feb 9 17:56:22 2018 by

LaTeXML
countable algebraic sets

countable algebraic sets

An

algebraic set

over an

uncountably infinite

base field

𝔽

(like the real or

complex numbers

) cannot be

countably infinite

.

Proof: Let

S

be a countably infinite subset of

𝔽

n

. By a

cardinality

argument

(see the attachment), there must exist a line such that the

projection

of this set to the line is

infinite

. Since the projection of an algebraic set to a linear subspace is an algebraic set, the projection of

S

to this line would be an

algebraic

subset of the line. However, an algebraic subset of a line is the locus of zeros of some

polynomial

, hence must be finite. Therefore,

S

could not be algebraic since that would lead to a

contradiction

.

Title

countable algebraic sets

Canonical name

CountableAlgebraicSets

Date of creation

2013-03-22 15:44:41

Last modified on

2013-03-22 15:44:41

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 14A10

Generated on Fri Feb 9 17:56:25 2018 by

LaTeXML
finite morphism

finite morphism

Affine schemes

Let

X

and

Y

be affine schemes, so that

X

=

Spec

⁡

A

and

Y

=

Spec

⁡

B

. Let

f

:

X

→

Y

be a

morphism

, so that it induces a

homomorphism

of rings

g

:

B

→

A

.

The homomorphism

g

makes

A

into a

B

-

algebra

. If

A

is
finitely-generated as a

B

-algebra, then

f

is said to be a morphism
of

finite type

.

If

A

is in fact

finitely generated

as a

B

-module, then

f

is said
to be a

finite

morphism.

For example, if

k

is a field, the scheme

𝔸

n

⁢

(

k

)

has a
natural morphism to

Spec

⁡

k

induced

by the

ring homomorphism

k

→

k

⁢

[

X

1

,

…

,

X

n

]

. This is a morphism of finite type, but if

n

>

0

then it is not a finite morphism.

On the other hand, if we take the affine scheme

Spec

⁡

k

⁢

[

X

,

Y

]

/

⟨

Y

2

-

X

3

-

X

⟩

, it has a natural morphism to

𝔸

1

given by the ring homomorphism

k

⁢

[

X

]

→

k

⁢

[

X

,

Y

]

/

⟨

Y

2

-

X

3

-

X

⟩

. Then this morphism is a finite
morphism. As a morphism of schemes, we see that every fiber is finite.

General schemes

Now, let

X

and

Y

be arbitrary schemes, and let

f

:

X

→

Y

be a morphism. We say that

f

is of

finite type

if there exist an
open cover of

Y

by affine schemes

{

U

i

}

and a finite open cover
of each

U

i

by affine schemes

{

V

i

⁢

j

}

such that

f

|

V

i

⁢

j

is a morphism of finite type for every

i

and

j

. We say that

f

is

finite

if there exists an open cover of

Y

by affine
schemes

{

U

i

}

such that each

inverse image

,

V

i

=

f

-

1

⁢

(

U

i

)

is
itself affine, and such that

f

|

V

i

is a finite morphism of affine
schemes.

Example.

Let

X

=

ℙ

1

⁢

(

k

)

and

Y

=

Spec

⁡

k

.
We cover

X

by two copies of

𝔸

1

and consider the natural
morphisms from each of these copies to

Spec

⁡

k

. Both of these
affine morphisms are of finite type, but are not finite. The

covering

morphisms patch together to give a morphism from

ℙ

1

to

Spec

⁡

k

. The overall morphism is of finite type, but again is not
finite.

References.

D. Eisenbud and J. Harris,

The Geometry of Schemes

, Springer.

Title

finite morphism

Canonical name

FiniteMorphism

Date of creation

2013-03-22 12:51:47

Last modified on

2013-03-22 12:51:47

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

9

Author

rmilson (146)

Entry type

Definition

Classification

msc 14A10

Classification

msc 14-00

Classification

msc 14A15

Defines

affine morphism

Defines

finite type

Generated on Fri Feb 9 17:56:27 2018 by

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