# 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 $C$.

 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