---
tag: Y2467453Q5
kind: concept
name: Algebraic K-theory of schemes
msc: ["19E08", 14C35]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_K-theory
---

## Summary

The algebraic K-theory of a scheme $X$ is a sequence of abelian groups $K_0(X), K_1(X), \ldots$ built from the category of [[vector bundles|P4DZM975BV]] on $X$.
The group $K_0(X)$ is the Grothendieck group of vector bundles, and the higher groups are defined by Quillen's Q-construction.
K-theory is the setting of the [[Grothendieck–Riemann–Roch theorem|3HKREN3Y3F]].
It is linked to [[Chow groups|EBEF6PEK4N]] by the Chern character and to motivic cohomology by spectral sequences.

## Explanations

### L1 human

The group $K_0(X)$ treats vector bundles like numbers: one can add them by direct sum, and a bundle built from two others by an extension counts as their sum.
For a point, vector bundles are vector spaces, and $K_0$ is $\Z$, measured by the dimension.
On the projective line, the bundles $\mathcal{O}(n)$ have both a rank and a degree, and $K_0(\mathbb{P}^1) \cong \Z^2$ records exactly these two numbers.
The higher groups $K_n$ capture finer information, such as units of rings for $K_1$.