Grothendieck group

Let S be an abelian semigroup. The Grothendieck group of S is K(S)=S×S/, where is the equivalence relationMathworldPlanetmath: (s,t)(u,v) if there exists rS such that s+v+r=t+u+r. This is indeed an abelian groupMathworldPlanetmath with zero elementMathworldPlanetmath (s,s) (any sS), inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath -(s,t)=(t,s) and addition given by (s,t)+(u,v)=(s+u,t+v). It is common to use the suggestive notation t-s for (t,s).

The Grothendieck group construction is a functorMathworldPlanetmath from the categoryMathworldPlanetmath of abelian semigroups to the category of abelian groups. A morphismMathworldPlanetmath f:ST induces a morphism K(f):K(S)K(T) which sends an element (s+,s-)K(S) to (f(s+),f(s-))K(T).

Example 1

Let (N,+) be the semigroup of natural numbersMathworldPlanetmath with composition given by addition. Then, K(N,+)=Z.

Example 2

Let (Z-{0},×) be the semigroup of non-zero integers with composition given by multiplication. Then, K(Z-{0},×)=(Q-{0},×).

Example 3

Let G be an abelian group, then K(G)G via the identification (g,h)g-h (or (g,h)gh-1 if G is multiplicative).

Let C be a (essentially small) symmetric monoidal category. Its Grothendieck group is K([C]), i.e. the Grothendieck group of the isomorphism classes of objects of C.

Title Grothendieck group
Canonical name GrothendieckGroup
Date of creation 2013-03-22 13:38:24
Last modified on 2013-03-22 13:38:24
Owner mhale (572)
Last modified by mhale (572)
Numerical id 11
Author mhale (572)
Entry type Definition
Classification msc 16E20
Classification msc 13D15
Classification msc 18F30
Synonym group completion
Related topic AlgebraicKTheory
Related topic KTheory
Related topic AlgebraicTopology
Related topic GrothendieckCategory