# Algebraic K-theory

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.

The functor ${K}_{\mathrm{0}}$

Let $R$ be a ring and denote by ${\mathrm{M}}_{\mathrm{\infty}}(R)$ the algebraic direct limit^{} of matrix algebras ${\mathrm{M}}_{n}(R)$ under the embeddings^{}
${\mathrm{M}}_{n}(R)\to {\mathrm{M}}_{n+1}(R):a\mapsto \left(\begin{array}{cc}\hfill a\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)$.
The zeroth K-group of $R$, ${K}_{0}(R)$, is the Grothendieck group (abelian group^{} of formal differences^{}) of idempotents^{} in ${\mathrm{M}}_{\mathrm{\infty}}(R)$ up to similarity transformations.
Let $p\in {\mathrm{M}}_{m}(R)$ and $q\in {\mathrm{M}}_{n}(R)$ 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 $p\oplus q=\mathrm{diag}(p,q)\in {\mathrm{M}}_{m+n}(R)$.
Equivalently, one can work with finitely generated projective modules over $R$.

The functor ${K}_{\mathrm{1}}$

Denote by ${\mathrm{GL}}_{\mathrm{\infty}}(R)$ the direct limit of general linear groups^{} ${\mathrm{GL}}_{n}(R)$ under the embeddings
${\mathrm{GL}}_{n}(R)\to {\mathrm{GL}}_{n+1}(R):g\mapsto \left(\begin{array}{cc}\hfill g\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)$.
Give ${\mathrm{GL}}_{\mathrm{\infty}}(R)$ the direct limit topology, i.e. a subset $U$ of ${\mathrm{GL}}_{\mathrm{\infty}}(R)$ is open if and only if
$U\cap {\mathrm{GL}}_{n}(R)$ is an open subset of ${\mathrm{GL}}_{n}(R)$, for all $n$.
The first K-group of $R$, ${K}_{1}(R)$, is the abelianisation of ${\mathrm{GL}}_{\mathrm{\infty}}(R)$, i.e.

$${K}_{1}(R)={\mathrm{GL}}_{\mathrm{\infty}}(R)/[{\mathrm{GL}}_{\mathrm{\infty}}(R),{\mathrm{GL}}_{\mathrm{\infty}}(R)].$$ |

Note that this is the same as ${K}_{1}(R)={H}_{1}({\mathrm{GL}}_{\mathrm{\infty}}(R),\mathbb{Z})$, the first group homology group (with integer coefficients).

The functor ${K}_{\mathrm{2}}$

Let ${\mathrm{E}}_{n}(R)$ be the elementary subgroup^{} of ${\mathrm{GL}}_{n}(R)$.
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 ${\mathrm{E}}_{\mathrm{\infty}}(R)$ the direct limit of the ${\mathrm{E}}_{n}(R)$ using the construction above (note ${\mathrm{E}}_{\mathrm{\infty}}(R)$ is a subgroup of ${\mathrm{GL}}_{\mathrm{\infty}}(R)$).
The second K-group of $R$, ${K}_{2}(R)$, is the second group homology group (with integer coefficients) of ${\mathrm{E}}_{\mathrm{\infty}}(R)$,

$${K}_{2}(R)={H}_{2}({\mathrm{E}}_{\mathrm{\infty}}(R),\mathbb{Z}).$$ |

Higher K-functors

Higher K-groups are defined using the Quillen plus construction,

$${K}_{n}^{\mathrm{alg}}(R)={\pi}_{n}(B{\mathrm{GL}}_{\mathrm{\infty}}{(R)}^{+}),$$ | (1) |

where $B{\mathrm{GL}}_{\mathrm{\infty}}(R)$ is the classifying space^{} of ${\mathrm{GL}}_{\mathrm{\infty}}(R)$.

Rough sketch of suspension:

$$\mathrm{\Sigma}R=\mathrm{\Sigma}\mathbb{Z}{\otimes}_{\mathbb{Z}}R$$ | (2) |

where $\mathrm{\Sigma}\mathbb{Z}=C\mathbb{Z}/J\mathbb{Z}$. The cone, $C\mathbb{Z}$, 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 $J\mathbb{Z}$ consists of those matrices that have only finitely many non-trivial coefficients.

$${K}_{i}(R)\cong {K}_{i+1}(\mathrm{\Sigma}R)$$ | (3) |

Algebraic K-theory has a product structure,

$${K}_{i}(R)\otimes {K}_{j}(S)\to {K}_{i+j}(R\otimes S).$$ | (4) |

## References

- 1 H. Inassaridze, Algebraic K-theory. Kluwer Academic Publishers, 1994.
- 2 Jean-Louis Loday, Cyclic Homology. Springer-Verlag, 1992.

Title | Algebraic K-theory |
---|---|

Canonical name | AlgebraicKtheory |

Date of creation | 2013-03-22 13:31:32 |

Last modified on | 2013-03-22 13:31:32 |

Owner | mhale (572) |

Last modified by | mhale (572) |

Numerical id | 10 |

Author | mhale (572) |

Entry type | Topic |

Classification | msc 19-00 |

Classification | msc 18F25 |

Related topic | KTheory |

Related topic | GrothendieckGroup |

Related topic | StableIsomorphism |