Grothendieck group

In mathematics, the Grothendieck group, or group of differences, of a commutative monoid M is a certain abelian group. This abelian group is constructed from M in the most universal way, in the sense that any abelian group containing a homomorphic image of M will also contain a homomorphic image of the Grothendieck group of M. The Grothendieck group construction takes its name from a specific case in category theory, introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K-theory. This specific case is the monoid of isomorphism classes of objects of an abelian category, with the direct sum as its operation.

Grothendieck group of a commutative monoid

Motivation

Given a commutative monoid

M

{\displaystyle M}

, "the most general" abelian group

K

{\displaystyle K}

that arises from

M

{\displaystyle M}

is to be constructed by introducing inverse elements to all elements of

M

{\displaystyle M}

. Such an abelian group

K

{\displaystyle K}

always exists; it is called the Grothendieck group of

M

{\displaystyle M}

. It is characterized by a certain universal property and can also be concretely constructed from

M

{\displaystyle M}

.

If

M

{\displaystyle M}

does not have the cancellation property (that is, there exists

a

,

b

{\displaystyle a,b}

and

c

{\displaystyle c}

in

M

{\displaystyle M}

such that

a

≠

b

{\displaystyle a\neq b}

and

a

c

=

b

c

{\displaystyle ac=bc}

), then the Grothendieck group

K

{\displaystyle K}

cannot contain

M

{\displaystyle M}

. In particular, in the case of a monoid operation denoted multiplicatively that has a zero element satisfying

0

⋅

x

=

0

{\displaystyle 0\cdot x=0}

for every

x

∈

M

,

{\displaystyle x\in M,}

the Grothendieck group must be the trivial group (group with only one element), since one must have

x

=

1

⋅

x

=

(

0

−

1

⋅

0

)

⋅

x

=

0

−

1

⋅

(

0

⋅

x

)

=

0

−

1

⋅

0

=

0

{\displaystyle x=1\cdot x=(0^{-1}\cdot 0)\cdot x=0^{-1}\cdot (0\cdot x)=0^{-1}\cdot 0=0}

for every

x

{\displaystyle x}

.

Universal property

Let M be a commutative monoid. Its Grothendieck group is an abelian group K with a monoid homomorphism

i

:

M

→

K

{\displaystyle i\colon M\to K}

satisfying the following universal property: for any monoid homomorphism

f

:

M

→

A

{\displaystyle f\colon M\to A}

from M to an abelian group A, there is a unique group homomorphism

g

:

K

→

A

{\displaystyle g\colon K\to A}

such that

f

=

g

∘

i

.