Group completion
In mathematics, the Grothendieck group construction in abstract algebra constructs an abelian group from a commutative monoid 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 the more general construction in category theory, introduced by Alexander Grothendieck in his fundamental work of the mid-1950s that resulted in the development of K-theory, which led to his proof of the Grothendieck-Riemann-Roch theorem. This article treats both constructions.
Given a commutative monoid M, we want to construct "the most general" abelian group K that arises from M by introducing additive inverses. Such an abelian group K always exists; it is called the Grothendieck group of M. It is characterized by a certain universal property and can also be concretely constructed from M.
Let M be a commutative monoid. Its Grothendieck group K is an abelian group with the following universal property: There exists a monoid homomorphism
such that for any monoid homomorphism
from the commutative monoid M to an abelian group A, there is a unique group homomorphism
such that
This expresses the fact that any abelian group A that contains a homomorphic image of M will also contain a homomorphic image of K, K being the "most general" abelian group containing a homomorphic image of M.
To construct the Grothendieck group K of a commutative monoid M, one forms the Cartesian product
(The two coordinates are meant to represent a positive part and a negative part: (m1, m2) is meant to correspond to the element m1 − m2 in K.)
Addition on MxM is defined coordinate-wise:
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 (the element k is necessary because the cancellation law does not hold in all monoids). The equivalence class of the element (m1, m2) is denoted by [(m1, m2)]. We define K to be the set of equivalence classes. Since the addition operation on M×M is compatible with our equivalence relation, we obtain an addition on K, and K becomes an abelian group. The identity element of K is [(0, 0)], and the inverse of [(m1, m2)] is [(m2, m1)]. The homomorphism i : M→K sends the element m to [(m, 0)].
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