# Grothendieck group

Let $S$ be an abelian semigroup. The Grothendieck group of $S$ is $K(S)=S\times S/\mathord{\sim}$, where $\sim$ is the equivalence relation: $(s,t)\sim(u,v)$ if there exists $r\in S$ such that $s+v+r=t+u+r$. This is indeed an abelian group with zero element $(s,s)$ (any $s\in S$), inverse $-(s,t)=(t,s)$ 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 functor from the category of abelian semigroups to the category of abelian groups. A morphism $f\colon S\to T$ induces a morphism $K(f)\colon K(S)\to K(T)$ which sends an element $(s^{+},s^{-})\in K(S)$ to $(f(s^{+}),f(s^{-}))\in K(T)$.

###### Example 1

Let $(\mathbb{N},+)$ be the semigroup of natural numbers with composition given by addition. Then, $K(\mathbb{N},+)=\mathbb{Z}$.

###### Example 2

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

###### Example 3

Let $G$ be an abelian group, then $K(G)\cong G$ via the identification $(g,h)\leftrightarrow g-h$ (or $(g,h)\leftrightarrow gh^{{-1}}$ 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 $C$.

Type of Math Object:
Definition
Major Section:
Reference

## Mathematics Subject Classification

16E20 no label found13D15 no label found18F30 Grothendieck groups

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## Info

Owner: mhale
Added: 2003-05-22 - 02:04
Author(s): mhale

## Versions

(v11) by mhale 2013-03-22
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