---
tag: 5GVFZWMWC8
kind: concept
name: Algebraic K-theory
msc: ["16E20", 19D06]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_K-theory
same_as:
  wikidata: Q2553675
  wikipedia: Algebraic K-theory
---

## Summary

Algebraic K-theory attaches to a [[ring|0B1261B5ED]] $R$ a sequence of abelian groups $K_n(R)$.
The group $K_0(R)$ is the [[Grothendieck group|5FDMY7PDYY]] of finitely generated [[projective modules|BFFA03GE3V]], and $K_1(R)$ is the abelianization of the infinite general linear group.
Daniel Quillen defined the higher groups by homotopy theory, for instance through the plus construction.
The groups encode deep arithmetic information, such as class groups, units and special values of zeta functions.

## Explanations

### L1 human

Algebraic K-theory measures a ring through its modules and its invertible matrices.
The group $K_0$ counts projective modules, with the rule that a direct sum becomes a sum.
For a field, projective modules are vector spaces, and $K_0$ just records the dimension, so it is $\Z$.
The group $K_1$ remembers invertible matrices up to elementary row operations; for a field this is the determinant, so $K_1$ is the group of nonzero elements.
The higher groups are harder to compute, and they connect algebra with topology and number theory.