Grothendieck Group - Explicit Construction

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 .