aarjaneiro/planetmath huggingface dataset:
prime ideal

prime ideal

Let

R

be a ring. A two-sided

proper ideal

𝔭

of a ring

R

is called a

prime ideal

if the following

equivalent

conditions are met:

1.

If

I

and

J

are

left ideals

and the

product of ideals

I

⁢

J

satisfies

I

⁢

J

⊂

𝔭

, then

I

⊂

𝔭

or

J

⊂

𝔭

.

2.

If

I

and

J

are

right ideals

with

I

⁢

J

⊂

𝔭

, then

I

⊂

𝔭

or

J

⊂

𝔭

.

3.

If

I

and

J

are

two-sided ideals

with

I

⁢

J

⊂

𝔭

, then

I

⊂

𝔭

or

J

⊂

𝔭

.

4.

If

x

and

y

are elements of

R

with

x

⁢

R

⁢

y

⊂

𝔭

, then

x

∈

𝔭

or

y

∈

𝔭

.

R

/

𝔭

is a

prime ring

if and only if

𝔭

is a prime ideal. When

R

is

commutative

with

identity

, a proper ideal

𝔭

of

R

is prime if and only if for any

a

,

b

∈

R

, if

a

⋅

b

∈

𝔭

then either

a

∈

𝔭

or

b

∈

𝔭

. One also has in this case that

𝔭

⊂

R

is prime if and only if the

quotient ring

R

/

𝔭

is an

integral domain

.

Title

prime ideal

Canonical name

PrimeIdeal

Date of creation

2013-03-22 11:50:54

Last modified on

2013-03-22 11:50:54

Owner

djao (24)

Last modified by

djao (24)

Numerical id

15

Author

djao (24)

Entry type

Definition

Classification

msc 16D99

Classification

msc 13C99

Related topic

MaximalIdeal

Related topic

Ideal

Related topic

PrimeElement

Generated on Fri Feb 9 18:40:09 2018 by

LaTeXML
socle

socle

Let

M

be a left (or right)

R

-module. The

socle

of

M

, denoted by

Soc

⁡

(

M

)

, is the sum of all simple (or minimal)

submodules

of

M

. If

M

has no simple submodules, then

Soc

⁡

(

M

)

:=

(

0

)

.

A left (or right)

R

-module

M

is

semisimple

iff

Soc

⁡

(

M

)

=

M

.

Title

socle

Canonical

name

Socle

Date of creation

2013-03-22 14:21:26

Last modified on

2013-03-22 14:21:26

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16D99

Related topic

SemisimpleGroup

Generated on Fri Feb 9 18:40:11 2018 by

LaTeXML
construction of an injective resolution

construction of an injective resolution

The

category

of modules has enough

injectives

.
Let

M

be a module, and let

I

0

be an

injective module

such that

0

⟶

M

⟶

I

0

is exact. Then, let

M

0

be the image of

M

in

I

0

, and

construct

the factor module

I

0

/

M

0

. Then, since the category of modules has enough injectives, we can find a module

I

1

such that

0

⟶

I

0

/

M

0

⟶

ϕ

0

I

1

is exact.

ϕ

0

induces a

homomorphism

ϕ

:

I

0

⟶

I

1

, whose kernel is

M

0

. We thus have an

exact sequence

0

⟶

M

⟶

I

0

⟶

I

1

.

One can continue this process to construct injective modules

I

n

for any

n

∈

ℤ

(the

resolution

may terminate:

I

m

=

0

for some

N

∈

ℤ

with all

m

>

N

).

Title

construction of an

injective resolution

Canonical name

ConstructionOfAnInjectiveResolution

Date of creation

2013-03-22 17:11:02

Last modified on

2013-03-22 17:11:02

Owner

guffin (12505)

Last modified by

guffin (12505)

Numerical id

5

Author

guffin (12505)

Entry type

Derivation

Classification

msc 16E05

Generated on Fri Feb 9 18:40:13 2018 by

LaTeXML
flat resolution

flat resolution

Let

M

be a module. A

flat resolution

of

M

is an

exact sequence

of the form

⋯

→

F

n

→

F

n

-

1

→

⋯

→

F

1

→

F

0

→

M

→

0

where each

F

n

is a

flat module

.

Title

flat resolution

Canonical name

FlatResolution

Date of creation

2013-03-22 12:09:53

Last modified on

2013-03-22 12:09:53

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

6

Author

antizeus (11)

Entry type

Definition

Classification

msc 16E05

Generated on Fri Feb 9 18:40:14 2018 by

LaTeXML
free resolution

free resolution

Let

M

be a module. A

free resolution

of

M

is an

exact sequence

of the form

⋯

→

F

n

→

F

n

-

1

→

⋯

→

F

1

→

F

0

→

M

→

0

where each

F

n

is a

free module

.

Title

free resolution

Canonical name

FreeResolution

Date of creation

2013-03-22 12:09:48

Last modified on

2013-03-22 12:09:48

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

6

Author

antizeus (11)

Entry type

Definition

Classification

msc 16E05

Generated on Fri Feb 9 18:40:16 2018 by

LaTeXML
injective resolution

injective resolution

Let

M

be a module. An

injective resolution

of

M

is an

exact sequence

of the form

0

→

M

→

Q

0

→

Q

1

→

⋯

→

Q

n

-

1

→

Q

n

→

⋯

where each

Q

n

is an

injective module

.

Title

injective resolution

Canonical name

InjectiveResolution

Date of creation

2013-03-22 12:09:56

Last modified on

2013-03-22 12:09:56

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

6

Author

antizeus (11)

Entry type

Definition

Classification

msc 16E05

Generated on Fri Feb 9 18:40:18 2018 by

LaTeXML
n-chain

n

-chain

Let

X

be a

topological space

and let

K

be a

simplicial approximation

to

X

. An

n

-chain

on

X

is a finite formal sum of oriented

n

-simplices in

K

. The group of such chains is denoted by

C

n

⁢

(

X

)

and is called the

n

th

chain group

of

X

. In other words,

C

n

⁢

(

X

)

is the

free abelian group

generated by the oriented

n

-simplices in

K

.

We have defined chain groups for

simplicial homology

. Their definition is similar in

singular homology

and the

homology

of

CW complexes

. For example, if

Y

is a CW complex, then its

n

th chain group is the free abelian group on the cells of

Y

n

, the

n

-

skeleton

of

Y

.

The formal

boundary

of an oriented

n

-simplex

σ

=

(

v

0

,

…

,

v

n

)

is given by the alternating sum of the oriented

n

-simplices forming the topological boundary of

σ

, that is,

∂

n

⁡

(

σ

)

=

∑

j

=

0

n

(

-

1

)

j

⁢

(

v

0

,

…

,

v

j

-

1

,

v

j

+

1

,

…

,

v

n

)

.

The boundary of a

0

-simplex is

0

.

Since

n

-simplices form a basis for the chain group

C

n

⁢

(

X

)

, this extends to give a

group homomorphism

∂

n

:

C

n

⁢

(

X

)

→

C

n

-

1

⁢

(

X

)

, called the

boundary map

. An

n

-chain is

closed

if its boundary is 0 and

exact

if it is the boundary of some

(

n

+

1

)

-chain. Closed

n

-chains are also called

cycles

. Every exact

n

-chain is also closed. This implies that the sequence

\xymatrix

⁢

⋯

⁢

\ar

⁢

[

r

]

⁢

&

⁢

C

n

+

1

⁢

(

X

)

⁢

\ar

⁢

[

r

]

∂

n

+

1

⁢

&

⁢

C

n

⁢

(

X

)

⁢

\ar

⁢

[

r

]

∂

n

⁢

&

⁢

C

n

-

1

⁢

\ar

⁢

[

r

]

⁢

&

⁢

⋯

is a complex of free abelian groups. This complex is usually called the

chain complex

of

X

corresponding to the

simplicial complex

K

. Note that while the chain groups

C

n

⁢

(

X

)

depend on the choice of simplicial approximation

K

, the resulting

homology groups

H

n

⁢

(

X

)

=

ker

⁢

∂

n

im

⁢

∂

n

+

1

do not.

Title

n

-chain

Canonical name

Nchain

Date of creation

2013-03-22 13:46:20

Last modified on

2013-03-22 13:46:20

Owner

mps (409)

Last modified by

mps (409)

Numerical id

11

Author

mps (409)

Entry type

Definition

Classification

msc 16E05

Synonym

chain

Related topic

LongExactSequenceOfHomologyGroups

Defines

closed n-chain

Defines

exact n-chain

Defines

boundary map

Generated on Fri Feb 9 18:40:21 2018 by

LaTeXML
projective resolution

projective resolution

Let

M

be a module. A

projective resolution

of

M

is an

exact sequence

of the form

⋯

→

P

n

→

P

n

-

1

→

⋯

→

P

1

→

P

0

→

M

→

0

where each

P

n

is a

projective module

.

Title

projective resolution

Canonical name

ProjectiveResolution

Date of creation

2013-03-22 12:09:51

Last modified on

2013-03-22 12:09:51

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

6

Author

antizeus (11)

Entry type

Definition

Classification

msc 16E05

Generated on Fri Feb 9 18:40:23 2018 by

LaTeXML
short exact sequence

short exact sequence

Let

A

,

B

,

C

be objects in an

abelian category

. A
is an

exact sequence

of the form

0

→

A

→

B

→

C

→

0

.

Note that in this case,
the

homomorphism

A

→

B

must be a

monomorphism

,
and the homomorphism

B

→

C

must be an

epimorphism

.

Title

short exact sequence

Canonical

name

ShortExactSequence

Date of creation

2013-03-22 12:09:29

Last modified on

2013-03-22 12:09:29

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

7

Author

antizeus (11)

Entry type

Definition

Classification

msc 16E05

Related topic

CategoricalSequence

Generated on Fri Feb 9 18:40:25 2018 by

LaTeXML
split short exact sequence

split short exact sequence

In an

abelian category

, a

short exact sequence

0

→

A

→

f

B

→

g

C

→

0

is

split

if it

satisfies

the following

equivalent

conditions:

(a) there exists a

homomorphism

h

:

C

→

B

such that

g

⁢

h

=

1

C

;

(b) there exists a homomorphism

j

:

B

→

A

such that

j

⁢

f

=

1

A

;

(c)

B

is

isomorphic

to the

direct sum

A

⊕

C

.

In this case, we say that

h

and

j

are

backmaps

or

splitting backmaps

.

Title

split short exact sequence

Canonical name

SplitShortExactSequence

Date of creation

2013-03-22 12:09:32

Last modified on

2013-03-22 12:09:32

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

8

Author

antizeus (11)

Entry type

Definition

Classification

msc 16E05

Synonym

backmap

Synonym

splitting backmap

Generated on Fri Feb 9 18:40:27 2018 by

LaTeXML
tensor product of chain complexes

tensor product of chain complexes

Let

C

′

=

{

C

n

′

,

∂

n

′

}

and

C

′′

=

{

C

n

′′

,

∂

n

′′

}

be two

chain complexes

of

R

-modules, where

R

is a

commutative ring

with unity. Their

tensor product

C

′

⊗

R

C

′′

=

{

(

C

′

⊗

R

C

′′

)

n

,

∂

n

}

is the chain complex defined by

(

C

′

⊗

R

C

′′

)

n

=

⊕

i

+

j

=

n

(

C

i

′

⊗

R

C

j

′′

)

,

∂

n

(

t

i

′

⊗

R

s

j

′′

)

=

∂

i

′

(

t

i

′

)

⊗

R

s

j

′′

+

(

-

1

)

i

t

i

′

⊗

R

∂

j

′′

(

s

j

′′

)

,

∀

t

i

′

∈

C

i

′

,

s

j

′′

∈

C

j

′′

,

(

i

+

j

=

n

)

,

where

C

i

′

⊗

R

C

j

′′

denotes the tensor product (

http://planetmath.org/TensorProduct

) of

R

-modules

C

i

′

and

C

j

′′

.

Indeed, this defines a chain complex, because for each

t

i

′

⊗

R

s

j

′′

∈

C

i

′

⊗

R

C

j

′′

⊆

(

C

′

⊗

R

C

′′

)

i

+

j

we have

∂

i

+

j

-

1

⁡

∂

i

+

j

⁡

(

t

i

′

⊗

R

s

j

′′

)

=

∂

i

+

j

-

1

⁡

(

∂

i

′

⁡

(

t

i

′

)

⊗

R

s

j

′′

+

(

-

1

)

i

⁢

t

i

′

⊗

R

∂

j

′′

⁡

(

s

j

′′

)

)

=

=

(

-

1

)

i

-

1

⁢

∂

i

′

⁡

(

t

i

′

)

⊗

R

∂

j

′′

⁡

(

s

j

′′

)

+

(

-

1

)

i

⁢

∂

i

′

⁡

(

t

i

′

)

⊗

R

∂

j

′′

⁡

(

s

j

′′

)

=

0

,

thus

C

′

⊗

R

C

′′

is a chain complex.

Title

tensor product of chain complexes

Canonical name

TensorProductOfChainComplexes

Date of creation

2013-03-22 16:13:21

Last modified on

2013-03-22 16:13:21

Owner

Mazzu (14365)

Last modified by

Mazzu (14365)

Numerical id

13

Author

Mazzu (14365)

Entry type

Definition

Classification

msc 16E05

Classification

msc 18G35

Defines

tensor product of chain complexes

Generated on Fri Feb 9 18:40:29 2018 by

LaTeXML
weak dimension of a module

weak dimension of a module

Assume that

R

is a ring. We will consider right

R

-modules.

Definition 1.

We will say that an

R

-module

M

is of

weak dimension

at most

n

∈

ℕ

iff there exists a

short exact sequence

\xymatrix

⁢

0

⁢

\ar

⁢

[

r

]

⁢

&

⁢

F

n

⁢

\ar

⁢

[

r

]

⁢

&

⁢

F

n

-

1

⁢

\ar

⁢

[

r

]

⁢

&

⁢

⋯

⁢

\ar

⁢

[

r

]

⁢

&

⁢

F

1

⁢

\ar

⁢

[

r

]

⁢

&

⁢

F

0

⁢

\ar

⁢

[

r

]

⁢

&

⁢

M

⁢

\ar

⁢

[

r

]

⁢

&

⁢

0

such that each

F

i

is a

flat module

. In this case we write

wd

R

⁢

M

⩽

n

(also we say that

M

is of finite weak dimension). If such short exact sequence does not exist, then the weak dimension is defined as

infinity

,

wd

R

⁢

M

=

∞

.

Definition 2.

We will say that an

R

-module

M

is of weak dimension

n

∈

ℕ

iff

wd

R

⁢

M

⩽

n

but

wd

R

⁢

M

⩽̸

n

-

1

.

The weak dimension measures how far an

R

-module is from being flat. Let as gather some known facts about the weak dimension:

Proposition

1.

Assume that

M

is a right

R

-module. Then

wd

R

⁢

M

=

n

for some

n

∈

ℕ

if and only if for any left

R

-module

N

we have

Tor

n

+

1

R

⁢

(

M

,

N

)

=

0

and there exists a left

R

-module

N

′

such that

Tor

n

R

⁢

(

M

,

N

′

)

≠

0

,

where

Tor

denotes the Tor functor.

Since every

projective module

is flat, then we can state simple observation:

Proposition 2.

Assume that

M

is a right

R

-module. Then

wd

R

⁢

M

⩽

pd

R

⁢

M

,

where

pd

R

⁢

M

denotes the

projective dimension

of

M

.

Generally these two dimension may differ.

Title

weak dimension of a module

Canonical name

WeakDimensionOfAModule

Date of creation

2013-03-22 19:18:40

Last modified on

2013-03-22 19:18:40

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Derivation

Classification

msc 16E05

Generated on Fri Feb 9 18:40:31 2018 by

LaTeXML
weak global dimension

weak global dimension

Let

R

be a ring. The

(right)

weak global dimension

of

R

is defined as

w

.

gl

.

dim

⁢

R

=

sup

⁢

{

wd

R

⁢

M

|

M

⁢

is a right module

}

.

Unlike

global dimension

of

R

the

definition

of the weak global dimension is left/right

symmetric

. This follows from the fact that for every

left module

M

and right module

N

there is an

isomorphism

Tor

n

R

⁢

(

M

,

N

)

≃

Tor

n

R

⁢

(

N

,

M

)

.

Thus we simply say that

R

has the weak global dimension. Note that this does not hold for Ext functors, so (generally) the definition of global dimension is not left/right symmetric.

The following

proposition

is a simple

consequence

of the fact that every

projective module

is flat:

Proposition.

For any ring

R

we have

w

.

gl

.

dim

R

⩽

min

{

l

.

gl

.

dim

R

,

r

.

gl

.

dim

R

}

,

where

l

.

gl

.

dim

and

r

.

gl

.

dim

denote the left global dimension and right global dimension respectively.

Title

weak global dimension

Canonical

name

WeakGlobalDimension

Date of creation

2013-03-22 19:18:42

Last modified on

2013-03-22 19:18:42

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Definition

Classification

msc 16E05

Generated on Fri Feb 9 18:40:33 2018 by

LaTeXML
exact sequences for modules with finite projective dimension

exact sequences for modules with finite projective dimension

Proposition

.

Let

R

be a ring and

M

be a (left)

R

-module, such that

proj dim

⁢

(

M

)

=

n

<

∞

. If

0

→

K

→

P

n

→

⋯

→

P

0

→

M

→

0

is an

exact sequence

of

R

-modules, such that each

P

i

is projective, then

K

is projective.

Proof.

Since

proj dim

⁢

(

M

)

=

n

<

∞

, then there exists exact sequence of

R

-modules

0

→

P

n

′

→

⋯

→

P

0

′

→

M

→

0

,

Note that

sequences

P

n

→

⋯

→

P

0

→

M

→

0

;

P

n

′

→

⋯

→

P

0

→

M

→

0

,

are

projective resolutions

of

M

. Let

δ

:

P

n

→

P

n

-

1

and

β

:

P

n

′

→

P

n

-

1

′

be maps take from these

resolutions

. Then generalized Schanuel’s lemma implies that

ker

⁢

δ

and

ker

⁢

β

are

projectively equivalent

. But

ker

⁢

δ

≃

K

and

ker

⁢

β

=

0

. This means, that there are

projective modules

P

,

Q

such that

K

⊕

P

≃

Q

.

Therefore

K

is a

direct summand

of a

free module

(since

Q

is), which

completes

the proof.

□

Title

exact sequences for modules with finite projective dimension

Canonical name

ExactSequencesForModulesWithFiniteProjectiveDimension

Date of creation

2013-03-22 19:04:55

Last modified on

2013-03-22 19:04:55

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Corollary

Classification

msc 16E10

Classification

msc 18G20

Classification

msc 18G10

Generated on Fri Feb 9 18:40:35 2018 by

LaTeXML
projective dimension

projective dimension

Let

𝒜

be an

abelian category

and

M

∈

Ob

⁡

(

𝒜

)

such that a

projective resolution

of

M

exists:

\xymatrix

⁢

…

⁢

\ar

⁢

[

r

]

⁢

&

⁢

P

n

⁢

\ar

⁢

[

r

]

⁢

&

⁢

…

⁢

\ar

⁢

[

r

]

⁢

&

⁢

P

1

⁢

\ar

⁢

[

r

]

⁢

&

⁢

P

0

⁢

\ar

⁢

[

r

]

⁢

&

⁢

M

⁢

\ar

⁢

[

r

]

⁢

&

⁢

0

.

Among all the projective resolutions of

M

, consider the subset consisting of those projective resolutions that contain only a finite number of non-zero

projective objects

(there exists a non-negative integer

n

such that

P

i

=

0

for all

i

≥

n

). If such a subset is non-empty, then the

projective dimension

of

M

is defined to be the smallest number

d

such that

\xymatrix

⁢

0

⁢

\ar

⁢

[

r

]

⁢

&

⁢

P

d

⁢

\ar

⁢

[

r

]

⁢

&

⁢

…

⁢

\ar

⁢

[

r

]

⁢

&

⁢

P

1

⁢

\ar

⁢

[

r

]

⁢

&

⁢

P

0

⁢

\ar

⁢

[

r

]

⁢

&

⁢

M

⁢

\ar

⁢

[

r

]

⁢

&

⁢

0

.

We denote this by

pd

⁡

(

M

)

=

d

. If this subset is empty, then we define

pd

⁡

(

M

)

=

∞

.

Remarks

.

1.

In an abelian category having enough projectives, the projective dimension of an object always exists (whether it is finite or not).

2.

If

pd

⁡

(

M

)

=

d

and

\xymatrix

⁢

0

⁢

\ar

⁢

[

r

]

⁢

&

⁢

P

d

⁢

\ar

⁢

[

r

]

⁢

&

⁢

…

⁢

\ar

⁢

[

r

]

⁢

&

⁢

P

1

⁢

\ar

⁢

[

r

]

⁢

&

⁢

P

0

⁢

\ar

⁢

[

r

]

⁢

&

⁢

M

⁢

\ar

⁢

[

r

]

⁢

&

⁢

0

.

Then

P

i

≠

0

for all

0

≤

i

≤

d

.

3.

pd

⁡

(

M

)

=

0

iff

M

is a projective object.

4.

In the (abelian)

category

of left (right)

R

-modules, the projective dimension of a left (right)

R

-module

M

is denoted by

pd

R

⁡

(

M

)

.

Likewise, given an abelian category and a object

N

having at least one

injective resolution

. Then the

injective dimension

, denoted by

id

⁡

(

N

)

is the minimum number

d

such that

\xymatrix

⁢

0

⁢

\ar

⁢

[

r

]

⁢

&

⁢

N

⁢

\ar

⁢

[

r

]

⁢

&

⁢

I

0

⁢

\ar

⁢

[

r

]

⁢

&

⁢

I

1

⁢

\ar

⁢

[

r

]

⁢

&

⁢

…

⁢

\ar

⁢

[

r

]

⁢

&

⁢

I

d

⁢

\ar

⁢

[

r

]

⁢

&

⁢

0

,

if such an injective resolution exists. Otherwise, set

id

⁡

(

N

)

=

∞

. This is the dual notion of projective dimension.

Title

projective dimension

Canonical name

ProjectiveDimension

Date of creation

2013-03-22 14:50:56

Last modified on

2013-03-22 14:50:56

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16E10

Classification

msc 13D05

Classification

msc 18G20

Defines

injective dimension

Generated on Fri Feb 9 18:40:37 2018 by

LaTeXML
projective equivalence

projective equivalence

Let

R

be a ring with 1. Two

R

-modules

A

and

B

are said to be

projectively equivalent

A

∼

B

if there exist
two projective

R

-modules

P

and

Q

such that

A

⊕

P

≅

B

⊕

Q

.

Remarks.

1.

Projective equivalence is an

equivalence relation

.

2.

Any

projective module

is projectively equivalent to the

zero module

.

3.

(Schanuel’s Lemma)

. Given two

short exact sequences

:

\xymatrix

⁢

0

⁢

\ar

⁢

[

r

]

⁢

&

⁢

B

1

⁢

\ar

⁢

[

r

]

⁢

&

⁢

P

⁢

\ar

⁢

[

r

]

⁢

&

⁢

A

1

⁢

\ar

⁢

[

r

]

⁢

&

⁢

0

\xymatrix

⁢

0

⁢

\ar

⁢

[

r

]

⁢

&

⁢

B

2

⁢

\ar

⁢

[

r

]

⁢

&

⁢

Q

⁢

\ar

⁢

[

r

]

⁢

&

⁢

A

2

⁢

\ar

⁢

[

r

]

⁢

&

⁢

0

with

A

1

∼

A

2

, then

B

1

∼

B

2

.

4.

Schanuel’s Lemma can be generalized. Given two

projective resolutions

:

\xymatrix

⁢

…

⁢

\ar

⁢

[

r

]

p

3

⁢

&

⁢

P

2

⁢

\ar

⁢

[

r

]

p

2

⁢

&

⁢

P

1

⁢

\ar

⁢

[

r

]

p

1

⁢

&

⁢

P

0

⁢

\ar

⁢

[

r

]

p

0

⁢

&

⁢

A

1

⁢

\ar

⁢

[

r

]

⁢

&

⁢

0

\xymatrix

⁢

…

⁢

\ar

⁢

[

r

]

q

3

⁢

&

⁢

Q

2

⁢

\ar

⁢

[

r

]

q

2

⁢

&

⁢

Q

1

⁢

\ar

⁢

[

r

]

q

1

⁢

&

⁢

Q

0

⁢

\ar

⁢

[

r

]

q

0

⁢

&

⁢

A

2

⁢

\ar

⁢

[

r

]

⁢

&

⁢

0

with

A

1

∼

A

2

, then

Ker

⁡

(

p

n

)

∼

Ker

⁡

(

q

n

)

for all

n

≥

0

5.

The

concept

of projective equivalence between two modules can be generalized to any

abelian categories

having enough projectives.

Title

projective equivalence

Canonical name

ProjectiveEquivalence

Date of creation

2013-03-22 14:50:13

Last modified on

2013-03-22 14:50:13

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16E10

Classification

msc 18G20

Classification

msc 18G10

Defines

projectively equivalent

Defines

Schanuel’s Lemma

Generated on Fri Feb 9 18:40:39 2018 by

LaTeXML
Grothendieck group

Grothendieck group

Let

S

be an

abelian semigroup

.
The

Grothendieck group

of

S

is

K

⁢

(

S

)

=

S

×

S

/

∼

,
where

∼

is the

equivalence relation

:

(

s

,

t

)

∼

(

u

,

v

)

if there exists

r

∈

S

such that

s

+

v

+

r

=

t

+

u

+

r

.
This is indeed an

abelian group

with

zero element

(

s

,

s

)

(any

s

∈

S

),

inverse

-

(

s

,

t

)

=

(

t

,

s

)

and

addition

given by

(

s

,

t

)

+

(

u

,

v

)

=

(

s

+

u

,

t

+

v

)

.
It is common to use the suggestive notation

t

-

s

for

(

t

,

s

)

.

The Grothendieck group construction is a

functor

from the

category

of abelian semigroups to the category of abelian groups.
A

morphism

f

:

S

→

T

induces a morphism

K

⁢

(

f

)

:

K

⁢

(

S

)

→

K

⁢

(

T

)

which sends an

element

(

s

+

,

s

-

)

∈

K

⁢

(

S

)

to

(

f

⁢

(

s

+

)

,

f

⁢

(

s

-

)

)

∈

K

⁢

(

T

)

.

Example 1

Let

(

N

,

+

)

be the

semigroup

of

natural numbers

with

composition

given by addition.
Then,

K

⁢

(

N

,

+

)

=

Z

.

Example 2

Let

(

Z

-

{

0

}

,

×

)

be the semigroup of non-zero integers with composition given by multiplication.
Then,

K

⁢

(

Z

-

{

0

}

,

×

)

=

(

Q

-

{

0

}

,

×

)

.

Example 3

Let

G

be an abelian group, then

K

⁢

(

G

)

≅

G

via the identification

(

g

,

h

)

↔

g

-

h

(or

(

g

,

h

)

↔

g

⁢

h

-

1

if

G

is multiplicative).

Let

C

be a (

essentially small

)

symmetric monoidal category

.
Its Grothendieck group is

K

⁢

(

[

C

]

)

,
i.e. the Grothendieck group of the isomorphism classes of objects of

C

.

Title

Grothendieck group

Canonical name

GrothendieckGroup

Date of creation

2013-03-22 13:38:24

Last modified on

2013-03-22 13:38:24

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

11

Author

mhale (572)

Entry type

Definition

Classification

msc 16E20

Classification

msc 13D15

Classification

msc 18F30

Synonym

group completion

Related topic

AlgebraicKTheory

Related topic

KTheory

Related topic

AlgebraicTopology

Related topic

GrothendieckCategory

Generated on Fri Feb 9 18:40:41 2018 by

LaTeXML