Editing K-theory (section)

==Grothendieck completion==
{{Main|Grothendieck group}}
The Grothendieck completion of an [[Monoid|abelian monoid]] into an abelian group is a necessary ingredient for defining K-theory since all definitions start by constructing an abelian monoid from a suitable category and turning it into an abelian group through this universal construction. Given an abelian monoid <math>(A,+')</math> let <math>\sim</math> be the relation on <math>A^2 = A \times A</math> defined by

:<math>(a_1,a_2) \sim (b_1,b_2)</math>

if there exists a <math>c\in A</math> such that <math>a_1 +' b_2 +' c = a_2 +' b_1 +' c.</math> Then, the set <math>G(A) = A^2/\sim</math> has the structure of a [[Group (mathematics)|group]] <math>(G(A),+)</math> where:

:<math> [(a_1,a_2)] + [(b_1,b_2)] = [(a_1+' b_1,a_2+' b_2)].</math>

Equivalence classes in this group should be thought of as formal differences of elements in the abelian monoid. This group <math>(G(A),+)</math> is also associated with a monoid homomorphism <math>i : A \to G(A)</math> given by <math>a \mapsto [(a, 0)],</math> which has a [[Grothendieck group#Universal property|certain universal property]].

To get a better understanding of this group, consider some [[equivalence class]]es of the abelian monoid <math>(A,+)</math>. Here we will denote the identity element of <math>A</math> by <math>0</math> so that <math>[(0,0)]</math> will be the identity element of <math>(G(A),+).</math> First, <math>(0,0) \sim (n,n)</math> for any <math>n\in A</math> since we can set <math>c = 0</math> and apply the equation from the equivalence relation to get <math>n = n.</math> This implies

:<math>[(a,b)] + [(b,a)] = [(a+b,a+b)] = [(0,0)]</math>

hence we have an additive inverse <math>[(b,a)]</math> for each <math>[(a,b)] \in G(A)</math>. This should give us the hint that we should be thinking of the equivalence classes <math>[(a,b)]</math> as formal differences <math>a-b.</math> Another useful observation is the invariance of equivalence classes under scaling:

:<math>(a,b) \sim (a+k,b+k)</math> for any <math>k \in A.</math>

The Grothendieck completion can be viewed as a [[functor]] <math>G:\mathbf{AbMon}\to\mathbf{AbGrp},</math> and it has the property that it is left adjoint to the corresponding [[forgetful functor]] <math>U:\mathbf{AbGrp}\to\mathbf{AbMon}.</math> That means that, given a morphism <math>\phi:A \to U(B)</math> of an abelian monoid <math>A</math> to the underlying abelian monoid of an abelian group <math>B,</math> there exists a unique abelian group morphism <math>G(A) \to B.</math>

=== Example for natural numbers ===
An illustrative example to look at is the Grothendieck completion of <math>\N</math>. We can see that <math>G((\N,+)) = (\Z,+).</math> For any pair <math>(a,b)</math> we can find a minimal representative <math>(a',b')</math> by using the invariance under scaling. For example, we can see from the scaling invariance that

:<math>(4,6) \sim (3,5) \sim (2,4) \sim (1,3) \sim (0,2)</math>

In general, if <math>k := \min\{a,b\}</math> then

:<math>(a,b) \sim (a-k,b-k)</math> which is of the form <math>(c,0)</math> or <math>(0,d).</math>

This shows that we should think of the <math>(a,0)</math> as positive integers and the <math>(0,b)</math> as negative integers.