---
tag: 6SX85ZKBNJ
kind: definition
name: Algebraic K-theory
msc: ["16E20", 19D06]
defines: 5GVFZWMWC8
uses: [0B1261B5ED, BFFA03GE3V, 5FDMY7PDYY, 0FCHE902D7, WA796ZCNRX, P976WYEFMB]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_K-theory
---

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity.

1. $K_0(R)$ is the [[Grothendieck group|5FDMY7PDYY]] of the commutative monoid of isomorphism classes of finitely generated [[projective|BFFA03GE3V]] left $R$-modules under direct sum.
2. Let $GL(R) = \bigcup_{n \ge 1} GL_n(R)$, where $GL_n(R)$ is the [[general linear group|0FCHE902D7]] and $GL_n(R) \subseteq GL_{n+1}(R)$ through $g \mapsto \begin{pmatrix} g & 0 \\ 0 & 1 \end{pmatrix}$.
Then $K_1(R) = GL(R)/[GL(R), GL(R)]$.
3. For $n \ge 1$, $K_n(R) = \pi_n(BGL(R)^+)$, the $n$-th [[homotopy group|P976WYEFMB]] of the plus construction of the [[classifying space|WA796ZCNRX]] $BGL(R)$ with respect to the commutator subgroup $[GL(R), GL(R)]$.

The *algebraic K-theory* of $R$ is the sequence of abelian groups $K_n(R)$, $n \ge 0$.

## Remarks

For $n = 1$ clause 3 gives back clause 2, since $\pi_1(BGL(R)^+) = GL(R)/[GL(R), GL(R)]$.
By Whitehead's lemma, $[GL(R), GL(R)]$ is the subgroup generated by elementary matrices.
Example: for a field $F$, every finitely generated module is free, so $K_0(F) \cong \Z$ by the dimension.
The [[determinant|00KDZRW4C0]] gives $K_1(F) \cong F^\times$, because the elementary matrices generate $SL(F)$.
Caution: replacing projective modules by all finitely generated modules gives a different group, $G_0(R)$.
For $R = k[\varepsilon]/(\varepsilon^2)$ over a field $k$, $K_0(R) \cong \Z$ is generated by $[R]$, $G_0(R) \cong \Z$ is generated by $[k]$, and $[R] = 2[k]$.