Algebraic K-theoryAlgebraicKTheory2003-03-20 10:37:372006-02-24 14:56:03Topic\usepackage{amssymb}
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\newcommand*{\defn}[1]{\textbf{#1}}Algebraic K-theory is a series of functors on the category of rings.
Broadly speaking, it classifies ring invariants, i.e.\ ring properties that are Morita invariant.
\textbf{The functor $K_0$}
Let $R$ be a ring and denote by $\Matrix{\infty}{R}$ the algebraic direct limit of matrix algebras $\Matrix{n}{R}$ under the embeddings
$\Matrix{n}{R} \to \Matrix{n+1}{R} : a \mapsto \left(\begin{array}{cc} a & 0 \\ 0 & 0 \end{array}\right)$.
The zeroth K-group of $R$, $K_0(R)$, is the Grothendieck group (abelian group of formal differences) of idempotents in $\Matrix{\infty}{R}$ up to similarity transformations.
Let $p \in \Matrix{m}{R}$ and $q \in \Matrix{n}{R}$ be two idempotents.
The sum of their equivalence classes $[p]$ and $[q]$ is the equivalence class of their direct sum:
$[p]+[q] = [p \oplus q]$ where $p \oplus q = \mathrm{diag}(p,q) \in \Matrix{m+n}{R}$.
Equivalently, one can work with finitely generated projective modules over $R$.
\textbf{The functor $K_1$}
Denote by $\GLgrp_\infty(R)$ the direct limit of general linear groups $\GLgrp_n(R)$ under the embeddings
$\GLgrp_n(R) \to \GLgrp_{n+1}(R) : g \mapsto \left(\begin{array}{cc} g & 0 \\ 0 & 1 \end{array}\right)$.
Give $\GLgrp_\infty(R)$ the direct limit topology, i.e.\
a subset $U$ of $\GLgrp_\infty(R)$ is open if and only if
$U \cap \GLgrp_n(R)$ is an open subset of $\GLgrp_n(R)$, for all $n$.
The first K-group of $R$, $K_1(R)$, is the abelianisation of $\GLgrp_\infty(R)$, i.e.\
\[
K_1(R) = \GLgrp_\infty(R)/[\GLgrp_\infty(R),\GLgrp_\infty(R)].
\]
Note that this is the same as $K_1(R) = H_1(\GLgrp_\infty(R), \Zset)$,
the first group homology group (with integer coefficients).
\textbf{The functor $K_2$}
Let $\Egrp_n(R)$ be the elementary subgroup of $\GLgrp_n(R)$.
That is, the group generated by the elementary $n\times n$ matrices $e_{ij}(r)$, $r\in R$,
where $e_{ij}(r)$ is the matrix with ones on the diagonals, the value $r$ in row $i$, column $j$
and zeros elsewhere.
Denote by $\Egrp_\infty(R)$ the direct limit of the $\Egrp_n(R)$ using the construction above (note $\Egrp_\infty(R)$ is a subgroup of $\GLgrp_\infty(R)$).
The second K-group of $R$, $K_2(R)$, is the second group homology group (with integer coefficients) of $\Egrp_\infty(R)$,
\[
K_2(R) = H_2(\Egrp_\infty(R), \Zset).
\]
\textbf{Higher K-functors}
Higher K-groups are defined using the Quillen plus construction,
\begin{equation}
K^{\mathrm{alg}}_n(R) = \pi_n(B\GLgrp_\infty(R)^+),
\end{equation}
where $B\GLgrp_\infty(R)$ is the classifying space of $\GLgrp_\infty(R)$.
Rough sketch of suspension:
\begin{equation}
\Sigma R = \Sigma\Zset \otimes_\Zset R
\end{equation}
where $\Sigma\Zset = C\Zset/J\Zset$.
The cone, $C\Zset$, is the set of infinite matrices with integral coefficients
that have a finite number of non-trivial elements on each row and column.
The ideal $J\Zset$ consists of those matrices that have only finitely many
non-trivial coefficients.
\begin{equation}
K_i(R) \cong K_{i+1}(\Sigma R)
\end{equation}
Algebraic K-theory has a product structure,
\begin{equation}
K_i(R) \otimes K_j(S) \to K_{i+j}(R \otimes S).
\end{equation}
\begin{thebibliography}{10}
\bibitem{Inassaridze}
H. Inassaridze, {\em Algebraic K-theory}.
\newblock Kluwer Academic Publishers, 1994.
\bibitem{Loday}
Jean-Louis Loday, {\em Cyclic Homology}.
\newblock Springer-Verlag, 1992.
\end{thebibliography}