aarjaneiro/planetmath huggingface dataset:
abelian variety

abelian variety

Definition 1

.

An

abelian variety

over a field

k

is a proper

group scheme

over

Spec

⁡

k

that is a

variety

.

This extremely terse definition needs some further explanation.

Proposition 1

.

The group law on an abelian variety is

commutative

.

This implies that for every ring

R

, the

R

-points of an abelian variety form an

abelian group

.

Proposition 2

.

An abelian variety is projective.

If

C

is a curve, then the Jacobian of

C

is an abelian variety. This example motivated the development of the theory of abelian varieties, and many properties of curves are best understood by looking at the Jacobian.

If

E

is an

elliptic curve

, then

E

is an abelian variety (and in fact

E

is naturally

isomorphic

to its Jacobian).

See Mumford’s excellent book

Abelian Varieties

. The

bibliography for algebraic geometry

has details and other books.

Title

abelian variety

Canonical name

AbelianVariety

Date of creation

2013-03-22 14:17:17

Last modified on

2013-03-22 14:17:17

Owner

archibal (4430)

Last modified by

archibal (4430)

Numerical id

6

Author

archibal (4430)

Entry type

Definition

Classification

msc 14K99

Generated on Fri Feb 9 17:59:31 2018 by

LaTeXML
general linear group scheme

general linear group scheme

Definition 1

Fix a

positive integer

n

. We define the

general linear group scheme

GL

n

as the

affine scheme

defined by

ℤ

⁢

[

Y

,

X

11

,

…

,

X

1

⁢

n

,

…

,

X

n

⁢

1

,

…

,

X

n

⁢

n

]

/

⟨

Y

⁢

det

⁡

(

X

11

⋯

X

1

⁢

n

⋮

⋱

⋮

X

n

⁢

1

⋯

X

n

⁢

n

)

-

1

⟩

Observe that if

R

is any commutative ring, as usual (

http://planetmath.org/ExampleOfFunctorOfPointsOfAScheme

) with schemes, an

R

-point of

GL

n

is given by specifying, for each

i

and

j

, an element

r

i

⁢

j

that is the image of

X

i

⁢

j

, and by specifying one other element

r

such that

r

⁢

det

⁡

(

r

11

⋯

r

1

⁢

n

⋮

⋱

⋮

r

n

⁢

1

⋯

r

n

⁢

n

)

=

1

.

In other words, an

R

-point of

GL

n

is an invertible matrix with entries in

R

.

As usual with schemes, we denote the

R

-points of

GL

n

by

GL

n

⁡

(

R

)

; we see that this notion does not lead to confusion, since it is exactly what is meant by the usual usage of this notation (see entry

General Linear Group

).

Title

general linear group scheme

Canonical name

GeneralLinearGroupScheme

Date of creation

2013-03-22 14:11:16

Last modified on

2013-03-22 14:11:16

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

7

Author

alozano (2414)

Entry type

Example

Classification

msc 14K99

Classification

msc 14A15

Classification

msc 14L10

Classification

msc 20G15

Related topic

GeneralLinearGroup

Generated on Fri Feb 9 17:59:33 2018 by

LaTeXML
group scheme

group scheme

A

group scheme

is a

group object

in the

category

of schemes. Similarly, if

S

is a scheme, a

group scheme over

S

is a group object in the category of schemes over

S

.

As usual with schemes, the points of a group scheme are not the whole story. For example, a group scheme may have only one point over its field of definition and yet not be trivial. The points of the underlying

topological space

do not form a group under the obvious choice for a group law.

We can view a group scheme

G

as a “group

machine

”: given a ring

R

, the set of

R

-points of

G

forms a group. If

S

is a scheme that is not affine, we can nevertheless interpret

G

as a family of groups fibred over

S

.

Title

group scheme

Canonical name

GroupScheme

Date of creation

2013-03-22 14:11:13

Last modified on

2013-03-22 14:11:13

Owner

archibal (4430)

Last modified by

archibal (4430)

Numerical id

4

Author

archibal (4430)

Entry type

Definition

Classification

msc 14K99

Classification

msc 14A15

Classification

msc 14L10

Classification

msc 20G15

Related topic

Group

Related topic

GroupVariety

Related topic

Category

Related topic

GroupObject

Related topic

GroupSchemeOfMultiplicativeUnits

Related topic

VarietyOfGroups

Generated on Fri Feb 9 17:59:35 2018 by

LaTeXML
ideal

ideal

Let

R

be a ring. A

left ideal

(resp.,

right ideal

)

I

of

R

is a nonempty subset

I

⊂

R

such that:

•

a

-

b

∈

I

for all

a

,

b

∈

I

•

r

⋅

a

∈

I

(resp.

a

⋅

r

∈

I

) for all

a

∈

I

and

r

∈

R

A

two-sided ideal

is a left ideal

I

which is also a right ideal. If

R

is a

commutative ring

, then these three notions of ideal are

equivalent

. Usually, the word “ideal” by itself means two-sided ideal.

The name “ideal” comes from the study of

number theory

. When the failure of

unique factorization

in

number fields

was first noticed, one of the solutions was to work with so-called “

ideal numbers

” in which unique factorization did hold. These “ideal numbers” were in fact ideals, and in

Dedekind domains

, unique factorization of ideals does indeed hold. The term “ideal number” is no longer used; the term “ideal” has replaced and generalized it.

Title

ideal

Canonical name

Ideal

Date of creation

2013-03-22 11:49:27

Last modified on

2013-03-22 11:49:27

Owner

djao (24)

Last modified by

djao (24)

Numerical id

18

Author

djao (24)

Entry type

Definition

Classification

msc 14K99

Classification

msc 16D25

Classification

msc 11N80

Classification

msc 13A15

Classification

msc 54C05

Classification

msc 54C08

Classification

msc 54J05

Classification

msc 54D99

Classification

msc 54E05

Classification

msc 54E15

Classification

msc 54E17

Related topic

Subring

Related topic

PrimeIdeal

Defines

left ideal

Defines

right ideal

Defines

2-sided ideal

Defines

two-sided ideal

Generated on Fri Feb 9 17:59:37 2018 by

LaTeXML
regular ideal

regular ideal

An ideal

𝔞

of a ring

R

is called a , iff

𝔞

a

regular element

of

R

.

Proposition

.

If

m

is a

positive integer

, then the only

regular ideal

in the

residue class ring

ℤ

m

is the

unit ideal

(

1

)

.

Proof.

The ring

ℤ

m

is a

principal ideal ring

.  Let

(

n

)

be any regular ideal of the ring

ℤ

m

.  Then

n

can not be

zero divisor

, since otherwise there would be a non-zero element

r

of

ℤ

m

such that

n

⁢

r

=

0

and thus every element

s

⁢

n

of the

principal ideal

would satisfy

(

s

⁢

n

)

⁢

r

=

s

⁢

(

n

⁢

r

)

=

s

⁢

0

=

0

.  So,

n

is a regular element of

ℤ

m

and therefore we have

gcd

⁡

(

m

,

n

)

=

1

.  Then, according to Bézout’s lemma (

http://planetmath.org/BezoutsLemma

), there are such

integers

x

and

y

that

1

=

x

⁢

m

+

y

⁢

n

.  This

equation

gives the

congruence

1

≡

y

⁢

n

(

mod

m

)

,  i.e.

1

=

y

⁢

n

in the ring

ℤ

m

.  With

1

the principal ideal

(

n

)

contains all elements of

ℤ

m

, which means that

(

n

)

=

ℤ

m

=

(

1

)

.

Note.

The above notion of “regular ideal” is used in most books concerning ideals of

commutative rings

, e.g. [1].  There is also a different notion of “regular ideal” mentioned in [2] (p. 179):  Let

I

be an ideal of the commutative ring

R

with

non-zero unity

.  This ideal is called

regular

, if the

quotient ring

R

/

I

is a

regular ring

, in other words, if for each

a

∈

R

there exists an element

b

∈

R

such that

a

2

⁢

b

-

a

∈

I

.

References

1

M. Larsen and P. McCarthy:

“

Multiplicative

theory of ideals

”.  Academic Press. New York (1971).

2

D. M. Burton:

“

A first course in rings and ideals

”.  Addison-Wesley. Reading, Massachusetts (1970).

Title

regular ideal

Canonical

name

RegularIdeal

Date of creation

2013-03-22 15:43:05

Last modified on

2013-03-22 15:43:05

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Definition

Classification

msc 14K99

Classification

msc 16D25

Classification

msc 11N80

Classification

msc 13A15

Related topic

QuasiRegularIdeal

Related topic

QuasiRegularity

Generated on Fri Feb 9 17:59:39 2018 by

LaTeXML
zero ideal

zero ideal

The subset

{

0

}

of a ring

R

is the least

two-sided ideal

of

R

.  As a

principal ideal

, it is often denoted by

(

0

)

and called the

zero ideal

.

The zero ideal is the

identity element

in the

addition of ideals

and the

absorbing element

in the

multiplication

of ideals (

http://planetmath.org/ProductOfIdeals

).  The

quotient ring

R

/

(

0

)

is trivially

isomorphic

to

R

.

By the entry

quotient ring modulo prime ideal

, (0) is a

prime ideal

if and only if

R

in an

integral domain

.

Title

zero ideal

Canonical name

ZeroIdeal1

Date of creation

2013-03-22 18:44:40

Last modified on

2013-03-22 18:44:40

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Definition

Classification

msc 14K99

Classification

msc 16D25

Classification

msc 11N80

Classification

msc 13A15

Related topic

MinimalPrimeIdeal

Related topic

PrimeRing

Related topic

ZeroModule

Generated on Fri Feb 9 17:59:41 2018 by

LaTeXML
group variety

group variety

Let

G

be a

variety

(an affine (

http://planetmath.org/AffineVariety

), projective (

http://planetmath.org/ProjectiveVariety

), or

quasi-projective variety

). We say

G

is a

group variety

if

G

is provided with morphisms of varieties:

μ

:

G

×

G

→

G

(

g

1

,

g

2

)

↦

g

1

⁢

g

2

,

ι

:

G

→

G

g

↦

g

-

1

,

and

ϵ

:

{

*

}

→

G

↦

e

,

and if these morphisms make the elements of

G

into a group.

In short,

G

should be a group object in the category of varieties. Examples include the

general linear group

of

dimension

n

on

k

and

elliptic curves

.

Group varieties that are actually projective are in fact

abelian groups

(although this is not

obvious

) and are called

abelian varieties

; their study is of

interest

to number theorists (among others).

Just as schemes generalize varieties,

group schemes

generalize group varieties. When dealing with situations in positive characteristic, or with families of group varieties, often they are more appropriate.

There is also a (not very closely related) concept in

group theory

of a “variety of groups (

http://planetmath.org/VarietyOfGroups

)”.

Title

group variety

Canonical

name

GroupVariety

Date of creation

2013-03-22 14:09:37

Last modified on

2013-03-22 14:09:37

Owner

archibal (4430)

Last modified by

archibal (4430)

Numerical id

4

Author

archibal (4430)

Entry type

Definition

Classification

msc 14L10

Classification

msc 14K99

Classification

msc 20G15

Related topic

AffineAlgebraicGroup

Related topic

GroupScheme

Related topic

VarietyOfGroups

Generated on Fri Feb 9 17:59:42 2018 by

LaTeXML
affine algebraic group

affine algebraic group

An

affine algebraic group

over a field

k

is

quasi-affine variety

G

(a locally closed subset of

affine space

) over

k

, which is a equipped with a group such that the

multiplication

map

m

:

G

×

G

→

G

and inverse map

i

:

G

→

G

are

algebraic

.

For example,

k

is an affine algebraic group over itself with the group law being

addition

, and as is

k

*

=

k

-

{

0

}

with the group law multiplication. Other common examples of affine algebraic groups are

GL

n

⁢

k

, the

general linear group

over

k

(identifying matrices with affine space) and any

algebraic torus

over

k

.

Title

affine algebraic group

Canonical

name

AffineAlgebraicGroup

Date of creation

2013-03-22 13:53:33

Last modified on

2013-03-22 13:53:33

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

7

Author

bwebste (988)

Entry type

Definition

Classification

msc 14L17

Related topic

GroupVariety

Generated on Fri Feb 9 17:59:44 2018 by

LaTeXML
algebraic torus

algebraic torus

Let

k

be a field. Then

k

*

, the

multiplicative group

of

k

is an

affine algebraic group

over

k

. An affine algebraic group of the form

(

k

*

)

n

is called an

algebraic torus

over

k

.

The name is to the fact that if

k

=

ℂ

, then an algebraic torus is the complexification of the standard torus

(

S

1

)

n

.

Title

algebraic torus

Canonical name

AlgebraicTorus

Date of creation

2013-03-22 13:53:27

Last modified on

2013-03-22 13:53:27

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

7

Author

bwebste (988)

Entry type

Definition

Classification

msc 14L17

Generated on Fri Feb 9 17:59:46 2018 by

LaTeXML
admissible ideals,, bound quiver and its algebra

admissible ideals,, bound quiver and its algebra

Assume, that

Q

is a quiver and

k

is a field. Let

k

⁢

Q

be the associated path

algebra

. Denote by

R

Q

the

two-sided ideal

in

k

⁢

Q

generated by all paths of length

1

, i.e. all arrows. This ideal is known as the

arrow ideal

.

It is easy to see, that for any

m

⩾

1

we have that

R

Q

m

is a two-sided ideal generated by all paths of length

m

. Note, that we have the following chain of ideals:

R

Q

2

⊇

R

Q

3

⊇

R

Q

4

⊇

⋯

Definition.

A two-sided ideal

I

in

k

⁢

Q

is said to be

admissible

if there exists

m

⩾

2

such that

R

Q

m

⊆

I

⊆

R

Q

2

.

If

I

is an admissible ideal in

k

⁢

Q

, then the pair

(

Q

,

I

)

is said to be a

bound quiver

and the

quotient algebra

k

⁢

Q

/

I

is called

bound quiver algebra

.

The idea behind this is to treat some paths in a quiver as

equivalent

. For example consider the following quiver

\xymatrix

⁢

&

⁢

2

⁢

\ar

⁢

[

d

⁢

r

]

b

⁢

&

⁢

1

⁢

\ar

⁢

[

r

⁢

r

]

c

⁢

\ar

⁢

[

d

⁢

r

]

e

⁢

\ar

⁢

[

u

⁢

r

]

a

⁢

&

⁢

&

⁢

3

⁢

&

⁢

4

⁢

\ar

⁢

[

u

⁢

r

]

f

Then the

ideal generated by

a

⁢

b

-

c

is not admissible (

a

⁢

b

-

c

∉

R

Q

2

) but an ideal generated by

a

⁢

b

-

e

⁢

f

is. We can see that this means that ,,walking” from

1

to

3

directly and through

2

is not the same, but walking in the same number of steps is.

Note, that in our case there is no path of length greater then

2

. In particular, for any

m

>

2

we have

R

Q

m

=

0

.

More generally, it can be easily checked, that if

Q

is a finite quiver without oriented cycles, then there exists

m

∈

ℕ

such that

R

Q

m

=

0

Title

admissible ideals,, bound quiver and its algebra

Canonical name

AdmissibleIdealsBoundQuiverAndItsAlgebra

Date of creation

2013-03-22 19:16:42

Last modified on

2013-03-22 19:16:42

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

6

Author

joking (16130)

Entry type

Definition

Classification

msc 14L24

Generated on Fri Feb 9 17:59:48 2018 by

LaTeXML
category of quivers is concrete

category of quivers is concrete

Let

𝒬

denote the

category

of all quivers and quiver

morphisms

with standard

composition

. If

Q

=

(

Q

0

,

Q

1

,

s

,

t

)

is a quiver, then we can associate with

Q

the set

S

⁢

(

Q

)

=

Q

0

⊔

Q

1

where ,,

⊔

” denotes the

disjoint union

of sets.

Furthermore, if

F

:

Q

→

Q

′

is a morphism of quivers, then

F

induces function

S

⁢

(

F

)

:

S

⁢

(

Q

)

→

S

⁢

(

Q

′

)

by putting

S

⁢

(

F

)

⁢

(

a

)

=

F

0

⁢

(

a

)

if

a

∈

Q

0

and

S

⁢

(

F

)

⁢

(

α

)

=

F

1

⁢

(

α

)

if

α

∈

Q

1

.

Proposition

.

The category

𝒬

together with

S

:

𝒬

→

𝒮

⁢

ℰ

⁢

𝒯

is a

concrete category

over the category of all sets

𝒮

⁢

ℰ

⁢

𝒯

.

Proof.

The fact that

S

is a

functor

we leave as a simple exercise. Now assume, that

F

,

G

:

Q

→

Q

′

are morphisms of quivers such that

S

⁢

(

F

)

=

S

⁢

(

G

)

. It follows, that for any vertex

a

∈

Q

0

and any arrow

α

∈

Q

1

we have

F

0

⁢

(

a

)

=

S

⁢

(

F

)

⁢

(

a

)

=

S

⁢

(

G

)

⁢

(

a

)

=

G

0

⁢

(

a

)

;

F

1

⁢

(

α

)

=

S

⁢

(

F

)

⁢

(

α

)

=

S

⁢

(

G

)

⁢

(

α

)

=

G

1

⁢

(

α

)

which clearly proves that

F

=

G

. This

completes

the proof.

□

Remark.

Note, that if

F

:

Q

→

Q

′

is a morphism of quivers, then

F

is

injective

in

(

𝒬

,

S

)

(see this entry (

http://planetmath.org/InjectiveAndSurjectiveMorphismsInConcreteCategories

) for details) if and only if both

F

0

,

F

1

are injective. The same holds if we replace word ,,injective” with ,,

surjective

”.

Title

category of quivers is concrete

Canonical

name

CategoryOfQuiversIsConcrete

Date of creation

2013-03-22 19:17:28

Last modified on

2013-03-22 19:17:28

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Theorem

Classification

msc 14L24

Generated on Fri Feb 9 17:59:50 2018 by

LaTeXML
characterization of isomorphisms of quivers

characterization of isomorphisms of quivers

Let

Q

=

(

Q

0

,

Q

1

,

s

,

t

)

and

Q

′

=

(

Q

0

′

,

Q

1

′

,

s

′

,

t

′

)

be quivers. Recall, that a

morphism

F

:

Q

→

Q

′

is an

isomorphism

if and only if there is a morphism

G

:

Q

′

→

Q

such that

F

⁢

G

=

Id

⁢

(

Q

′

)

and

G

⁢

F

=

Id

⁢

(

Q

)

, where

Id

⁢

(

Q

)

:

Q

→

Q

is given by

Id

⁢

(

Q

)

=

(

Id

⁢

(

Q

)

0

,

Id

⁢

(

Q

)

1

)

, where both

Id

⁢

(

Q

)

0

and

Id

⁢

(

Q

)

1

are the

identities

on

Q

0

,

Q

1

respectively.

Proposition

.

A morphism of quivers

F

:

Q

→

Q

′

is an isomorphism if and only if both

F

0

and

F

1

are bijctions.

Proof.

,,

⇒

” It follows from the

definition

of isomorphism that

F

0

⁢

G

0

=

Id

⁢

(

Q

′

)

0

and

G

0

⁢

F

0

=

Id

⁢

(

Q

)

0

for some

G

0

:

Q

0

′

→

Q

0

. Thus

F

0

is a

bijection

. The same

argument

is valid for

F

1

.

,,

⇐

” Assume that both

F

0

and

F

1

are bijections and define

G

:

Q

0

′

→

Q

0

and

H

:

Q

1

′

→

Q

1

by

G

=

F

0

-

1

,

H

=

F

1

-

1

.

Obviously

(

G

,

H

)

is ,,the

inverse

” of

F

in the sense, that the equalites for

compositions

hold. What is remain to prove is that

(

G

,

H

)

is a morphism of quivers. Let

α

∈

Q

1

′

. Then there exists an arrow

β

∈

Q

1

such that

F

1

⁢

(

β

)

=

α

.

Thus

H

⁢

(

α

)

=

β

.

Since

F

is a morphism of quivers, then

s

′

⁢

(

α

)

=

s

′

⁢

(

F

1

⁢

(

β

)

)

=

F

0

⁢

(

s

⁢

(

β

)

)

,

which implies that

G

⁢

(

s

′

⁢

(

α

)

)

=

s

⁢

(

β

)

=

s

⁢

(

H

⁢

(

α

)

)

.

The same arguments hold for the target function

t

, which

completes

the proof.

□

Title

characterization of isomorphisms of quivers

Canonical

name

CharacterizationOfIsomorphismsOfQuivers

Date of creation

2013-03-22 19:17:31

Last modified on

2013-03-22 19:17:31

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Theorem

Classification

msc 14L24

Generated on Fri Feb 9 17:59:52 2018 by

LaTeXML
First Isomorphism Theorem for quivers

First Isomorphism Theorem for quivers

Let

Q

=

(

Q

0

,

Q

1

,

s

,

t

)

and

Q

′

=

(

Q

0

′

,

Q

1

′

,

s

′

,

t

′

)

be quivers. Assume, that

F

:

Q

→

Q

′

is a

morphism

of quivers. Define an

equivalence relation

∼

on

Q

as follows: for any

a

,

b

∈

Q

0

and any

α

,

β

∈

Q

1

we have

a

∼

0

b

⁢

if and only if

⁢

F

0

⁢

(

a

)

=

F

0

⁢

(

b

)

;

α

∼

1

β

⁢

if and only if

⁢

F

1

⁢

(

α

)

=

F

1

⁢

(

β

)

.

It can be easily checked that

∼

=

(

∼

0

,

∼

1

)

is an equivalence relation on

Q

.

Using standard techniques we can prove the following:

First Isomorphism Theorem for quivers

.

The

mapping

F

¯

:

(

Q

/

∼

)

→

Im

(

F

)

(where on the left side we have the

quotient quiver

(

http://planetmath.org/QuotientQuiver

) and on the right side the image of a quiver (

http://planetmath.org/SubquiverAndImageOfAQuiver

)) given by

F

¯

0

⁢

(

[

a

]

)

=

F

0

⁢

(

a

)

,

F

¯

1

⁢

(

[

α

]

)

=

F

1

⁢

(

α

)

is an

isomorphism

of quivers.

Proof.

It easily follows from the definition of

∼

that

F

¯

is a well-defined morphism of quivers. Thus it is enough to show, that

F

¯

is both ,,onto” and ,,1-1” (in the sense that corresponding

components

of

F

¯

are).

1.

We will show, that

F

¯

is onto, i.e. both

F

¯

0

,

F

¯

1

are onto. Let

b

∈

Im

⁢

(

F

)

0

and

β

∈

Im

⁢

(

F

)

1

. By definition

F

0

⁢

(

a

)

=

b

,

F

1

⁢

(

α

)

=

β

for some

a

∈

Q

0

,

α

∈

Q

1

. It follows that

F

¯

0

⁢

(

[

a

]

)

=

b

,

F

¯

1

⁢

(

[

α

]

)

=

β

.

which

completes

this part.

2.

F

¯

is

injective

. Indeed, if

F

¯

0

⁢

(

[

a

]

)

=

F

¯

0

⁢

(

[

b

]

)

then

F

0

⁢

(

a

)

=

F

0

⁢

(

b

)

. But then

a

∼

0

b

and thus

[

a

]

=

[

b

]

. Analogously we prove the statement for

F

¯

1

.

This completes the proof.

□

Title

First Isomorphism Theorem for quivers

Canonical name

FirstIsomorphismTheoremForQuivers

Date of creation

2013-03-22 19:17:25

Last modified on

2013-03-22 19:17:25

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

5

Author

joking (16130)

Entry type

Definition

Classification

msc 14L24

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

LaTeXML
locally finite quiver

locally finite quiver

Let

Q

=

(

Q

0

,

Q

1

,

s

,

t

)

be a quiver, i.e.

Q

0

is a set of vertices,

Q

1

is a set of arrows and

s

,

t

:

Q

1

→

Q

0

are source and target

functions

.

Definition.

We will say that

Q

is

locally finite

iff for any vertex

a

∈

Q

0

there is a finite number of neighbours (

http://planetmath.org/PredecessorsAndSuccesorsInQuivers

) of

a

. Equivalently if there is a finite number of arrows ending in

a

and finite number of arrows starting from

a

.

Note that even when

Q

has a finite number of vertices, then

Q

is not necessarily locally finite. Consider the following example:

Q

=

(

{

*

}

,

ℕ

,

s

,

t

)

such that

s

⁢

(

n

)

=

t

⁢

(

n

)

=

*

for any

n

∈

ℕ

. In other words

Q

has one vertex and countably many arrows starting and ending at it. This quiver is not locally finite.

Every finite quiver, i.e. quiver with finite number of vertices and arrows is locally finite.

Title

locally finite quiver

Canonical name

LocallyFiniteQuiver

Date of creation

2013-03-22 19:17:51

Last modified on

2013-03-22 19:17:51

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Definition

Classification

msc 14L24

Generated on Fri Feb 9 17:59:57 2018 by

LaTeXML
morphisms between bound quivers

morphisms between bound quivers

Let

(

Q

,

I

)

and

(

Q

′

,

I

′

)

be bound quivers (

http://planetmath.org/AdmissibleIdealsBoundQuiverAndItsAlgebra

) over the same

base field

k

.

Definition.

A morphism

F

:

Q

→

Q

′

is said to be

bounded by

(

I

,

I

′

)

if the

induced

linear map (

http://planetmath.org/MorphismsOfPathAlgebrasInducedFromMorphismsOfQuivers

)

F

¯

:

k

⁢

Q

→

k

⁢

Q

′

is such that

F

¯

⁢

(

I

)

⊆

I

′

.

In this case we write

F

:

(

Q

,

I

)

→

(

Q

′

,

I

′

)

and we say that

F

is a

morphism of bound quivers

.

If

F

:

(

Q

,

I

)

→

(

Q

′

,

I

′

)

is a morphism of bound quivers, then

F

¯

:

k

⁢

Q

→

k

⁢

Q

′

induces a linear map

F

¯

¯

:

k

⁢

Q

/

I

→

k

⁢

Q

′

/

I

′

.

Furthermore, if

F

0

is injective, then

F

¯

is a

homomorphism

of

algebras

(see this entry (

http://planetmath.org/MorphismsOfPathAlgebrasInducedFromMorphismsOfQuivers

) for details) and thus

F

¯

¯

is a homormorphism of algebras.

Title

morphisms between bound quivers

Canonical name

MorphismsBetweenBoundQuivers

Date of creation

2013-03-22 19:17:07

Last modified on

2013-03-22 19:17:07

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

5

Author

joking (16130)

Entry type

Definition

Classification

msc 14L24

Generated on Fri Feb 9 18:00:01 2018 by

LaTeXML