Grothendieck group

Let $S$ be an abelian semigroup. The Grothendieck group of $S$ is $K(S)=S\times S/\mathrm{\sim }$, 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 $f:S\to T$ induces a morphism $K(f):K(S)\to K(T)$ which sends an element $(s^{+},s^{-})\in K(S)$ to $(f(s^{+}),f(s^{-}))\in K(T)$.

Example 1

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

Example 2

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

Example 3

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