aarjaneiro/planetmath huggingface dataset:
ind-variety

ind-variety

Let

𝕂

be a field. An

ind-variety

over

𝕂

is a set

X

along with a

filtration

:

X

0

⊂

X

1

⊂

⋯

⁢

X

n

⊂

⋯

such that

1.

X

=

⋃

j

≥

0

X

j

2.

Each

X

i

is a finite dimensional algebraic variety over

𝕂

3.

The

inclusions

i

j

:

X

j

→

X

j

+

1

are closed

embeddings

of algebraic varieties

The ring of

regular functions

on an ind-variety

X

is defined to be

𝕂

⁡

[

X

]

:=

lim

←

⁡

𝕂

⁡

[

X

j

]

where the limit is taken
with respect to the family of maps

{

i

j

*

:

𝕂

⁡

[

X

j

+

1

]

→

𝕂

⁡

[

X

j

]

}

j

≥

0

.

This ring is given the

structure

of a

topological ring

by letting each

𝕂

⁡

[

X

j

]

have the

discrete topology

and

𝕂

⁡

[

X

]

have the induced

inverse limit

topology

, i.e. the topology induced from the

canonical

inclusion

lim

←

⁡

𝕂

⁡

[

X

j

]

⊂

∏

j

𝕂

⁡

[

X

j

]

and the

product topology

on

∏

j

𝕂

⁡

[

X

j

]

.

An ind-variety is called

affine

(resp.

projective

) if each

X

j

is affine (resp. projective).

The notion of an ind-variety goes back to Igor Shafarevich in

[

3

]

and

[

4

]

.

Examples

Let

𝒦

:=

𝕂

⁡

(

(

t

)

)

be the ring of formal Laurant
series over

𝕂

and

𝒪

:=

𝕂

⁡

[

[

t

]

]

be its
ring of integers, the formal

Taylor series

. Let

V

=

𝕂

n

. Then the set

X

of

𝒪

-lattices (

𝒪

-

submodules

of

maximal

rank) in

V

⊗

𝕂

𝒦

is an example of a (non-finite
dimensional) projective ind-variety using the filtration

X

i

:=

{

L

∈

X

∣

t

i

⁢

L

0

⊂

L

⊂

t

-

i

⁢

L

0

,

dim

𝕂

⁡

L

/

t

i

⁢

L

0

=

i

⁢

n

}

where

L

0

:=

V

⊗

𝕂

𝒪

.

(cf.

[

1

]

section

11, or

[

2

]

appendix C part 7)

References

1

George Lusztig,

Singularities, character formulas,
and a q-analog of weight

multiplicities

, Astérisque 101-102 (1983), pp. 208-229.

2

Shrawan Kumar,

Kac-Moody Groups, their Flag Varieties and

Representation Theory

. Progress in Mathematics Vol. 204. Birkhauser, 2002.

3

Igor Shafarevich,

On some infinite-dimensional groups. II

Math USSR Izvestija 18 (1982), pp. 185 - 194.

4

Igor Shafarevich,

Letter to the editors: ”On some infinite-dimensional groups. II“

Izv. Ross. Akad. Nauk. Ser. Mat. 59 (1995), pp. 224 - 224.

Title

ind-variety

Canonical name

Indvariety

Date of creation

2013-03-22 15:30:56

Last modified on

2013-03-22 15:30:56

Owner

benjaminfjones (879)

Last modified by

benjaminfjones (879)

Numerical id

7

Author

benjaminfjones (879)

Entry type

Definition

Classification

msc 14A10

Classification

msc 14L15

Defines

ind-variety

Generated on Fri Feb 9 17:56:30 2018 by

LaTeXML
isomorphism of varieties

isomorphism of varieties

Definition 1

.

Let

V

1

and

V

2

be

algebraic varieties

. We say that

V

1

and

V

2

are

isomorphic

, and write

V

1

≅

V

2

, if there are

regular maps

ϕ

:

V

1

→

V

2

,

ψ

:

V

2

→

V

1

such that the

compositions

ψ

∘

ϕ

and

ϕ

∘

ψ

are the

identity maps

on

V

1

and

V

2

respectively.

Definition 2

.

Let

V

1

and

V

2

be

varieties

defined over a field

K

. We say that

V

1

/

K

and

V

2

/

K

are isomorphic over

K

if

V

1

and

V

2

are isomorphic as in

Definition

1 and the regular maps

ϕ

and

ψ

can be defined over

K

.

Title

isomorphism of varieties

Canonical

name

IsomorphismOfVarieties

Date of creation

2013-03-22 15:06:22

Last modified on

2013-03-22 15:06:22

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Definition

Classification

msc 14A10

Related topic

JInvariantClassifiesEllipticCurvesUpToIsomorphism

Generated on Fri Feb 9 17:56:31 2018 by

LaTeXML
lemma on projection of countable sets

lemma on projection of countable sets

Suppose

𝔽

is an infinite field and

S

is an infinite subset of

𝔽

n

. Then there exists a line

L

such that the

projection

of

S

on

L

is infinite.

Proof: This proof will proceed by an induction on

n

. The case

n

=

1

is trivial since a one-dimensional

linear space

is a line.

Consider two cases:

Case I: There exists a proper

subspace

of

𝔽

n

which contains an infinite number of points of

S

.

In this case, we can restrict attention to this subspace. By the
induction hypothesis, there exists a line in the subspace such that
the

projection of points

in the subspace to this line is already
infinite.

Case II: Every proper subspace of

𝔽

n

contains at most a
finite number of points of

S

.

In this case, any line will do. By definition, one constructs a
projection by dropping

hyperplanes

perpendicular

to the line

passing
through

the points of the set. Since each of these hyperplanes will
contain a finite number of elements of

S

, an infinite number of
hyperplanes will be needed to contain all the points of

S

, hence the
projection will be infinite.

Title

lemma on projection of countable sets

Canonical name

LemmaOnProjectionOfCountableSets

Date of creation

2013-03-22 15:44:53

Last modified on

2013-03-22 15:44:53

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

10

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 14A10

Generated on Fri Feb 9 17:56:33 2018 by

LaTeXML
Noetherian topological space

Noetherian topological space

A

topological space

X

is called if it

satisfies

the

descending chain condition

for

closed subsets

: for any

sequence

Y

1

⊇

Y

2

⊇

⋯

of closed subsets

Y

i

of

X

, there is an integer

m

such that

Y

m

=

Y

m

+

1

=

⋯

.

As a first example, note that all finite topological spaces are

Noetherian

.

There is a lot of interplay between the Noetherian condition and

compactness

:

•

Every

Noetherian topological space

is

quasi-compact

.

•

A

Hausdorff topological space

X

is Noetherian if and only if every

subspace

of

X

is

compact

. (i.e.

X

is hereditarily compact)

Note that if

R

is a

Noetherian ring

, then

Spec

⁢

(

R

)

, the

prime spectrum

of

R

, is a Noetherian topological space.

Example of a Noetherian topological space:

The space

𝔸

k

n

(affine

n

-space over a field

k

) under the

Zariski topology

is an example of a Noetherian topological space. By properties of the ideal of a subset of

𝔸

k

n

, we know that if

Y

1

⊇

Y

2

⊇

⋯

is a descending chain of Zariski-closed subsets, then

I

⁢

(

Y

1

)

⊆

I

⁢

(

Y

2

)

⊆

⋯

is an ascending chain of ideals of

k

⁢

[

x

1

,

…

,

x

n

]

.

Since

k

⁢

[

x

1

,

…

,

x

n

]

is a Noetherian ring, there exists an integer

m

such that

I

⁢

(

Y

m

)

=

I

⁢

(

Y

m

+

1

)

=

⋯

. But because we have a

one-to-one correspondence

between

radical ideals

of

k

⁢

[

x

1

,

…

,

x

n

]

and Zariski-closed sets in

𝔸

k

n

, we have

V

⁢

(

I

⁢

(

Y

i

)

)

=

Y

i

for all

i

. Hence

Y

m

=

Y

m

+

1

=

⋯

as required.

Title

Noetherian topological space

Canonical name

NoetherianTopologicalSpace

Date of creation

2013-03-22 13:03:33

Last modified on

2013-03-22 13:03:33

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

18

Author

mathcam (2727)

Entry type

Definition

Classification

msc 14A10

Related topic

Compact

Generated on Fri Feb 9 17:56:35 2018 by

LaTeXML
regular map

regular map

A

regular map

ϕ

:

k

n

→

k

m

between

affine spaces

over an algebraically closed field is merely one given by polynomials. That is, there are

m

polynomials

F

1

,

…

,

F

m

in

n

variables

such that the map is given by

ϕ

⁢

(

x

1

,

…

,

x

n

)

=

(

F

1

⁢

(

x

)

,

…

,

F

m

⁢

(

x

)

)

where

x

stands for the many

components

x

i

.

A regular map

ϕ

:

V

→

W

between

affine varieties

is one which is the

restriction

of a regular map between affine spaces. That is, if

V

⊂

k

n

and

W

⊂

k

m

, then there is a regular map

ψ

:

k

n

→

k

m

with

ψ

⁢

(

V

)

⊂

W

and

ϕ

=

ψ

|

V

. So, this is a map given by polynomials, whose image lies in the intended target.

A regular map between

algebraic varieties

is a locally regular map. That is

ϕ

:

V

→

W

is

regular

if around each point

x

there is an affine variety

V

x

and around each point

f

⁢

(

x

)

∈

W

there is an affine variety

W

f

⁢

(

x

)

with

ϕ

⁢

(

V

x

)

⊂

W

f

⁢

(

x

)

and such that the restriction

V

x

→

W

f

⁢

(

x

)

is a regular map of affine varieties.

Title

regular map

Canonical

name

RegularMap

Date of creation

2013-03-22 12:04:00

Last modified on

2013-03-22 12:04:00

Owner

nerdy2 (62)

Last modified by

nerdy2 (62)

Numerical id

6

Author

nerdy2 (62)

Entry type

Definition

Classification

msc 14A10

Synonym

regular morphism

Generated on Fri Feb 9 17:56:36 2018 by

LaTeXML
structure sheaf

structure sheaf

Let

X

be an

irreducible

algebraic variety

over a field

k

, together with the

Zariski topology

. Fix a point

x

∈

X

and let

U

⊂

X

be any affine

open subset

of

X

containing

x

. Define

𝔬

x

:=

{

f

/

g

∈

k

⁢

(

U

)

∣

f

,

g

∈

k

⁢

[

U

]

,

g

⁢

(

x

)

≠

0

}

,

where

k

⁢

[

U

]

is the

coordinate ring

of

U

and

k

⁢

(

U

)

is the

fraction field

of

k

⁢

[

U

]

. The ring

𝔬

x

is

independent

of the choice of affine open neighborhood

U

of

x

.

The

structure sheaf

on the

variety

X

is the sheaf of rings whose

sections

on any open subset

U

⊂

X

are given by

𝒪

X

⁢

(

U

)

:=

⋂

x

∈

U

𝔬

x

,

and where the

restriction

map for

V

⊂

U

is the

inclusion map

𝒪

X

⁢

(

U

)

↪

𝒪

X

⁢

(

V

)

.

There is an

equivalence of categories

under which an

affine variety

X

with its structure sheaf corresponds to the

prime spectrum

of the coordinate ring

k

⁢

[

X

]

. In fact, the

topological embedding

X

↪

Spec

⁡

(

k

⁢

[

X

]

)

gives rise to a

lattice

–preserving

bijection

1

1

Those who are fans of topos theory will recognize this map as an

isomorphism

of topos.

between the open sets of

X

and of

Spec

⁡

(

k

⁢

[

X

]

)

, and the sections of the structure sheaf on

X

are isomorphic to the sections of the sheaf

Spec

⁡

(

k

⁢

[

X

]

)

.

Title

structure sheaf

Canonical name

StructureSheaf

Date of creation

2013-03-22 12:38:20

Last modified on

2013-03-22 12:38:20

Owner

djao (24)

Last modified by

djao (24)

Numerical id

4

Author

djao (24)

Entry type

Definition

Classification

msc 14A10

Generated on Fri Feb 9 17:56:38 2018 by

LaTeXML
Zariski topology

Zariski topology

Let

𝔸

k

n

denote the

affine space

k

n

over a field

k

. The

Zariski topology

on

𝔸

k

n

is defined to be the topology whose closed sets are the sets

V

⁢

(

I

)

:=

{

x

∈

𝔸

k

n

∣

f

⁢

(

x

)

=

0

⁢

for all

⁢

f

∈

I

}

⊂

𝔸

k

n

,

where

I

⊂

k

⁢

[

X

1

,

…

,

X

n

]

is any ideal in the

polynomial ring

k

⁢

[

X

1

,

…

,

X

n

]

. For any

affine variety

V

⊂

𝔸

k

n

, the

Zariski topology

on

V

is defined to be the

subspace topology

induced on

V

as a subset of

𝔸

k

n

.

Let

ℙ

k

n

denote

n

–dimensional

projective space

over

k

. The

Zariski topology

on

ℙ

k

n

is defined to be the topology whose closed sets are the sets

V

⁢

(

I

)

:=

{

x

∈

ℙ

k

n

∣

f

⁢

(

x

)

=

0

⁢

for all

⁢

f

∈

I

}

⊂

ℙ

k

n

,

where

I

⊂

k

⁢

[

X

0

,

…

,

X

n

]

is any

homogeneous ideal

in the graded

k

–

algebra

k

⁢

[

X

0

,

…

,

X

n

]

. For any

projective variety

V

⊂

ℙ

k

n

, the

Zariski topology

on

V

is defined to be the subspace topology induced on

V

as a subset of

ℙ

k

n

.

The Zariski topology is the predominant topology used in the study of

algebraic geometry

. Every

regular morphism

of

varieties

is continuous in the Zariski topology (but not every continuous map in the Zariski topology is a regular morphism). In fact, the Zariski topology is the weakest topology on varieties making points in

𝔸

k

1

closed and regular morphisms continuous.

Title

Zariski topology

Canonical name

ZariskiTopology

Date of creation

2013-03-22 12:38:11

Last modified on

2013-03-22 12:38:11

Owner

djao (24)

Last modified by

djao (24)

Numerical id

4

Author

djao (24)

Entry type

Definition

Classification

msc 14A10

Related topic

PrimeSpectrum

Generated on Fri Feb 9 17:56:40 2018 by

LaTeXML
closed immersion

closed immersion

A

morphism of schemes

f

:

(

X

,

𝒪

X

)

⟶

(

Y

,

𝒪

Y

)

is a

closed immersion

if:

1.

As a map of

topological spaces

,

f

:

X

⟶

Y

is a

homeomorphism

from

X

into a

closed subset

of

Y

, and

2.

the

morphism of sheaves

𝒪

Y

⟶

𝒪

X

associated with

f

is an

epimorphism

(

http://planetmath.org/AbelianCategory

) in the category of sheaves.

This notion is the analog of the notion of closed immersion (

http://planetmath.org/Immersion

) in the

category

of

differential manifolds

.

Title

closed immersion

Canonical name

ClosedImmersion

Date of creation

2013-03-22 12:50:21

Last modified on

2013-03-22 12:50:21

Owner

djao (24)

Last modified by

djao (24)

Numerical id

6

Author

djao (24)

Entry type

Definition

Classification

msc 14A15

Generated on Fri Feb 9 17:56:42 2018 by

LaTeXML
coherent sheaf

coherent sheaf

Let

R

be a

ring with unity

, and

X

=

Spec

⁢

R

be its

prime spectrum

. Given an

R

-module

M

, one can define a

presheaf

on

X

by defining its

sections

on an

open set

U

to be

𝒪

X

⁢

(

U

)

⊗

R

M

. We call the

sheafification

of this

M

~

, and a sheaf of this form on

X

is called

quasi-coherent

. If

M

is a

finitely generated module

, then

M

~

is called coherent. A sheaf on an arbitrary scheme

X

is called (quasi-)coherent if it is (quasi-)coherent on each open affine subset of

X

.

Title

coherent sheaf

Canonical name

CoherentSheaf

Date of creation

2013-03-22 13:51:27

Last modified on

2013-03-22 13:51:27

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

11

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 14A15

Synonym

quasi-coherent sheaf

Generated on Fri Feb 9 17:56:43 2018 by

LaTeXML
example of fibre product

example of fibre product

Let

G

,

G

′

, and

H

be groups, and suppose we have

homomorphisms

f

:

G

→

H

and

f

′

:

G

′

→

H

. Then we can

construct

the fibre product

G

×

H

G

′

. It is the following group:

{

(

g

,

g

′

)

∈

G

×

G

′

⁢

such that

⁢

f

⁢

(

g

)

=

f

′

⁢

(

g

′

)

}

.

Observe that since

f

and

f

′

are homomorphisms, it is

closed under

the

group operations

.

Note also that the fibre product depends on the maps

f

and

F

′

, although the notation does not reflect this.

Title

example of fibre product

Canonical name

ExampleOfFibreProduct

Date of creation

2013-03-22 14:08:38

Last modified on

2013-03-22 14:08:38

Owner

archibal (4430)

Last modified by

archibal (4430)

Numerical id

4

Author

archibal (4430)

Entry type

Example

Classification

msc 14A15

Related topic

Group

Related topic

Homomorphism

Related topic

CartesianProduct

Generated on Fri Feb 9 17:56:45 2018 by

LaTeXML
fibre product

fibre product

Let

S

be a scheme, and let

i

:

X

⟶

S

and

j

:

Y

⟶

S

be schemes over

S

. A

fibre product

of

X

and

Y

over

S

is a scheme

X

×

S

Y

together with morphisms

p

:

X

×

S

Y

⟶

X

q

:

X

×

S

Y

⟶

Y

such that given any scheme

Z

with morphisms

x

:

Z

⟶

X

y

:

Z

⟶

Y

where

i

∘

x

=

j

∘

y

, there exists a unique morphism

(

x

,

y

)

:

Z

⟶

X

×

S

Y

making the diagram

Generated on Fri Feb 9 17:56:52 2018 by

LaTeXML
flat morphism

flat morphism

Let

f

:

X

→

Y

be a

morphism of schemes

. Then a sheaf

ℱ

of

𝒪

X

-modules is

flat over

Y

at a point

x

∈

X

if

ℱ

x

is a flat (

http://planetmath.org/FlatModule

)

𝒪

Y

,

f

⁢

(

x

)

-module by way of the map

f

♯

:

𝒪

Y

→

𝒪

X

associated to

f

.

The

morphism

f

itself is said to be

flat

if

𝒪

X

is flat over

Y

at every point of

X

.

This is the natural condition for

X

to form a “continuous family” over

Y

. That is, for each

y

∈

Y

, the fiber

X

y

of

f

over

y

is a scheme. We can consider

X

as a family of schemes parameterized by

Y

. If the morphism

f

is flat, then this family should be thought of as a “continuous family”. In particular, this means that certain cohomological invariants remain constant on the fibers of

X

.

References

1

Robin Hartshorne,

Algebraic
Geometry

, Springer–Verlag, 1977 (GTM

52

).

Title

flat morphism

Canonical name

FlatMorphism

Date of creation

2013-03-22 14:11:10

Last modified on

2013-03-22 14:11:10

Owner

archibal (4430)

Last modified by

archibal (4430)

Numerical id

4

Author

archibal (4430)

Entry type

Definition

Classification

msc 14A15

Synonym

flat

Related topic

Scheme

Related topic

EtaleMorphism

Defines

flat sheaf

Generated on Fri Feb 9 17:56:53 2018 by

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generically finite morphism

generically finite morphism

Let

f

:

X

→

Y

be a

morphism of schemes

. Assume

Y

is

irreducible

and let

y

be the generic point of

Y

. Then

f

is

generically finite

if

f

-

1

⁢

(

y

)

is finite.

Title

generically finite morphism

Canonical

name

GenericallyFiniteMorphism

Date of creation

2013-03-22 18:09:17

Last modified on

2013-03-22 18:09:17

Owner

neapol1s (9480)

Last modified by

neapol1s (9480)

Numerical id

6

Author

neapol1s (9480)

Entry type

Definition

Classification

msc 14A15

Defines

Generically Finite Morphism

Defines

Generically Finite

Generated on Fri Feb 9 17:56:55 2018 by

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group scheme of multiplicative units

group scheme of multiplicative units

Let

R

=

ℤ

⁢

[

X

,

Y

]

/

⟨

X

⁢

Y

-

1

⟩

. Then

Spec

⁡

R

is an

affine scheme

. The

natural homomorphism

ℤ

→

R

makes

R

into a scheme over

Spec

⁡

ℤ

, i.e. a

ℤ

-scheme.

What are the

ℤ

-points of

Spec

⁡

R

? Recall that an

S

-point of a scheme

X

is a morphism

S

→

X

; if we are working in the category of schemes over

Y

, then the morphism is expected to commute with the structure morphisms. So, here, we seek

homomorphisms

ℤ

⁢

[

X

,

Y

]

/

⟨

X

⁢

Y

-

1

⟩

→

ℤ

. Such a homomorphism must take

X

to an invertible element, and it must take

Y

to its

inverse

. Therefore there are two, one taking

X

to

1

and one taking

X

to

-

1

. One recognizes these as the

multiplicative

units of

ℤ

, and indeed if

S

is any ring, then the

S

-points of

Spec

⁡

R

are exactly the multiplicative units of

S

. For this reason, this scheme is often denoted

𝔾

m

. It is an example of a

group scheme

.

We can regard any morphism as a family of schemes, one for each fibre.
Since we have a morphism

𝔾

m

→

ℤ

, we can ask about the fibres of this morphism. If we select a point

x

of

Spec

⁡

ℤ

, we have two choices. Such a point must be a

prime ideal

of

ℤ

, and there are two kinds:

ideals generated by

a

prime number

, and the

zero ideal

. If we select a point

x

with

residue field

k

⁢

(

x

)

, then the fiber of this morphism will be

Spec

⁡

R

×

Spec

⁡

k

⁢

(

x

)

, which is the same as

Spec

⁡

R

⊗

k

⁢

(

x

)

. But looking at the definition of

R

, we see that this is

Spec

⁡

k

⁢

(

x

)

⁢

[

X

,

Y

]

/

⟨

X

⁢

Y

-

1

⟩

, which is just the scheme whose points are the nonzero elements of

k

⁢

(

x

)

.

In other words, we have a family of schemes, one in each

characteristic

. Of course, normally one wants a family to have some additional sort of smoothness condition, but this demonstrates that it is quite possible to have a family of schemes in different characteristics; sometimes one can deduce the behaviour in one characteristic from the behaviour in another. This approach can be useful, for example, when dealing with Hilbert

modular

varieties

.

Title

group scheme of multiplicative units

Canonical name

GroupSchemeOfMultiplicativeUnits

Date of creation

2013-03-22 14:09:01

Last modified on

2013-03-22 14:09:01

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Example

Classification

msc 14A15

Synonym

𝔾

m

Related topic

GroupScheme

Defines

group scheme of multiplicative units

Generated on Fri Feb 9 17:56:57 2018 by

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irreducible

irreducible

A subset

F

of a

topological space

X

is

reducible

if it can be written as a union

F

=

F

1

∪

F

2

of two closed

proper subsets

F

1

,

F

2

of

F

(closed in the

subspace topology

). That is,

F

is reducible if it can be written as a union

F

=

(

G

1

∩

F

)

∪

(

G

2

∩

F

)

where

G

1

,

G

2

are

closed subsets

of

X

, neither of which contains

F

.

A subset of a topological space is

irreducible

(or

hyperconnected

) if it is not reducible.

As an example, consider

{

(

x

,

y

)

∈

ℝ

2

:

x

⁢

y

=

0

}

with the subspace topology from

ℝ

2

. This space is a union of two lines

{

(

x

,

y

)

∈

ℝ

2

:

x

=

0

}

and

{

(

x

,

y

)

∈

ℝ

2

:

y

=

0

}

, which are proper closed subsets. So this space is reducible, and thus not irreducible.

Title

irreducible

Canonical name

Irreducible1

Date of creation

2013-03-22 12:03:30

Last modified on

2013-03-22 12:03:30

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

14

Author

mathcam (2727)

Entry type

Definition

Classification

msc 14A15

Classification

msc 14A10

Classification

msc 54B05

Related topic

IrreducibleComponent

Related topic

HyperconnectedSpace

Defines

reducible

Generated on Fri Feb 9 17:56:58 2018 by

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morphism of schemes induces a map of points

morphism of schemes induces a map of points

Let

f

:

X

→

Y

be a

morphism of schemes

over

S

, and let

T

be a particular scheme over

S

. Then

f

induces a natural function from the

S

-points of

X

to the

S

-points of

T

.

Recall that a

T

-point of

X

is a morphism

ϕ

:

T

→

X

. So examine the following diagram:

\xymatrix

T

\ar

[

d

r

r

]

ϕ

\ar

[

d

d

r

r

r

]

\ar

@

-

-

>

[

d

r

r

r

r

]

ψ

&

&

&

&

&

&

X

\ar

[

r

r

]

f

\ar

[

d

r

]

&

&

Y

\ar

[

d

l

]

&

&

&

S

&

Since all the schemes in question are

S

-schemes, the solid arrows all commute. The dashed arrow

ψ

we simply construct as

f

∘

ϕ

, making the whole diagram commute. The

ψ

is a

T

-point of

Y

.

Title

morphism of schemes induces a map of points

Canonical name

MorphismOfSchemesInducesAMapOfPoints

Date of creation

2013-03-22 14:11:02

Last modified on

2013-03-22 14:11:02

Owner

archibal (4430)

Last modified by

archibal (4430)

Numerical id

4

Author

archibal (4430)

Entry type

Result

Classification

msc 14A15

Generated on Fri Feb 9 17:57:02 2018 by

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