Grothendieck group GrothendieckGroup 2003-05-21 22:04:51 2008-11-20 01:23:58 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amsthm} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % making logically defined graphics %\usepackage{xypic} % my maths package \newcommand*{\Nset}{\mathbb{N}} \newcommand*{\Zset}{\mathbb{Z}} \newcommand*{\Qset}{\mathbb{Q}} \newcommand*{\Rset}{\mathbb{R}} \newcommand*{\Cset}{\mathbb{C}} \newcommand*{\Hset}{\mathbb{H}} \newcommand*{\Oset}{\mathbb{O}} \newcommand*{\Bset}{\mathbb{B}} \newcommand*{\Kset}{\mathbb{K}} \newcommand*{\Sset}{\mathbb{S}} \newcommand*{\Tset}{\mathbb{T}} \newcommand*{\GLgrp}{\mathrm{GL}} \newcommand*{\SLgrp}{\mathrm{SL}} \newcommand*{\Ogrp}{\mathrm{O}} \newcommand*{\SOgrp}{\mathrm{SO}} \newcommand*{\Ugrp}{\mathrm{U}} \newcommand*{\SUgrp}{\mathrm{SU}} \newcommand*{\e}{\mathop{\mathrm{e}}\nolimits} \newcommand*{\im}{\mathord{\mathrm{i}}} \newcommand*{\identity}{\mathord{\mathrm{1\!\!\!\:I}}} \newcommand*{\tr}{\mathop{\mathrm{tr}}} \newcommand*{\Tr}{\mathop{\mathrm{Tr}}} \renewcommand*{\d}{\mathrm{d}} \newcommand*{\deriv}[2]{\frac{\d #1}{\d #2}} \newcommand*{\pderiv}[2]{\frac{\partial #1}{\partial #2}} \newcommand*{\fderiv}[2]{\frac{\delta #1}{\delta #2}} % my noncommutative geometry package \newcommand*{\algebra}[1][A]{\mathord{\mathcal{#1}}} \newcommand*{\hilbert}[1][H]{\mathord{\mathcal{#1}}} \newcommand*{\hilbmod}[1][E]{\mathord{\mathcal{#1}}} \newcommand*{\Matrix}[2]{\mathord{\mathrm{M}_{#1}(#2)}} \newcommand*{\dixmier}{\mathop{\mathrm{Tr}_\omega}} \newcommand*{\Res}{\mathop{\mathrm{Res}}} \newcommand*{\Wres}{\mathop{\mathrm{Wres}}} \newcommand*{\Aut}{\mathop{\mathrm{Aut}}\nolimits} \newcommand*{\Inn}{\mathop{\mathrm{Inn}}\nolimits} \newcommand*{\Out}{\mathop{\mathrm{Out}}\nolimits} \newcommand*{\Diff}{\mathop{\mathrm{Diff}}\nolimits} \newcommand*{\Ker}{\mathop{\mathrm{Ker}}\nolimits} \newcommand*{\Coker}{\mathop{\mathrm{Coker}}\nolimits} \newcommand*{\Img}{\mathop{\mathrm{Im}}\nolimits} \newcommand*{\End}{\mathop{\mathrm{End}}\nolimits} \newcommand*{\spin}{\mathop{\mathrm{spin}}\nolimits} \newcommand*{\Ind}{\mathop{\mathrm{Ind}}\nolimits} \newcommand*{\KK}{\mathit{KK}} \newcommand*{\HH}{\mathit{HH}} \newcommand*{\HC}{\mathit{HC}} \newcommand*{\ch}{\mathop{\mathrm{ch}}\nolimits} % my category theory package \newcommand*{\mathcat}[1]{\mathord{\mathbf{#1}}} \newcommand*{\id}{\mathrm{id}} \newcommand*{\op}{\mathrm{op}} \newcommand*{\boxprod}{\mathbin{\square}} % my environments \newtheoremstyle{inlinedefn}{}{0pt}{}{}{\bfseries}{.}{0.5em}{} \theoremstyle{inlinedefn} \newtheorem{definition}{Definition} \newtheoremstyle{break}{\baselineskip}{\baselineskip}{\itshape}{}{\bfseries}{}{\newline}{} \theoremstyle{break} \newtheorem{example}{Example} % misc commands \newcommand*{\defn}[1]{\textbf{#1}} Let $S$ be an abelian semigroup. The \textbf{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 $(f(s^+),f(s^-)) \in K(T)$. \begin{example} Let $(\Nset,+)$ be the semigroup of natural numbers with composition given by addition. Then, $K(\Nset,+) = \Zset$. \end{example} \begin{example} Let $(\Zset-\lbrace 0 \rbrace,\times)$ be the semigroup of non-zero integers with composition given by multiplication. Then, $K(\Zset-\lbrace 0 \rbrace,\times) = (\Qset-\lbrace 0 \rbrace,\times)$. \end{example} \begin{example} 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). \end{example} 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 $C$.