Grothendieck Groups

Keywords: K-theory, Category Theory

Universal property

In its simplest form, the Grothendieck group of a commutative monoid is the universal way of making that monoid into an abelian group. Let M be a commutative monoid. Its Grothendieck group N should have the following universal property: There exists a monoid homomorphism

i:M→N

such that for any monoid homomorphism

f:M→A

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

g:N→A

such that

f=gi.

In the language of category theory, the functor that sends a commutative monoid M to its Grothendieck group N is left adjoint to the forgetful functor from the category of abelian groups to the category of commutative monoids.

Explicit construction

To construct the Grothendieck group of a commutative monoid M, one forms the Cartesian product

M×M.

The two coordinates are meant to represent a positive part and a negative part:

(m, n)

is meant to correspond to

m − n.

Addition is defined coordinate-wise:

(m₁, m₂) + (n₁, n₂) = (m₁ + n₁, m₂ + n₂).

Next we define an equivalence relation on M×M. We say that (m₁, m₂) is equivalent to (n₁, n₂) if, for some element k of M, m₁ + n₂ + k = m₂ + n₁ + k. It is easy to check that the addition operation is compatible with the equivalence relation. The identity element is now any element of the form (m, m), and the inverse of (m₁, m₂) is (m₂, m₁).

In this form, the Grothendieck group is the fundamental construction of K-theory. The group K₀(M) of a manifold M is defined to be the Grothendieck group of the commutative monoid of all isomorphism classes of vector bundles of finite rank on M with the monoid operation given by direct sum. The zeroth algebraic K group K₀(R) of a ring R is the Grothendieck group of the monoid consisting of isomorphism classes of projective modules over R, with the monoid operation given by the direct sum.

The Grothendieck group can also be constructed using generators and relations: denoting by (Z(M),+') the free abelian group generated by the set M, the Grothendieck group is the quotient of Z(M) by the subgroup generated by [math]\displaystyle{ \{x+'y-'(x+y)\mid x,y\in M\} }[/math].