nLab
algebraic K-theory

Idea

Algebraic K-theory is the study of K-theory of categories more general than that of (bounded chain complexes of) vector bundles on a topological space, which is the topic of topological K-theory.

In its simple form algebraic K-theory provides tools for computing the Grothendieck group of suitable categories. In its more refined form it studies the K-theory spectrum assigned to these categories. Crucuial tools for this go by the name Q-construction? and Waldhausen S-construction.

Types of categories for which a theory of algebraic K-theory exist include notable the notions

• Quillen exact category

• abelian category

• Waldhausen category,

• triangulated category.

Concrete examples of interest include for instance

• the category of finitely generated projective objects over a unital $k$-algebra,

• the category of coherent sheaves over a noetherian scheme,

• the category of locally free sheaves over a scheme,

and the like.

When one talks about the algebraic $K$-theory of rings, one means the algebraic $K$-theory of the corresponding category of (one-sided) finitely-generated projective modules.

A K-theory should be given by a sequence of functors $K_i$ from some class of categories as above to abelian groups having some similarity to derived functors and cohomology theory for spaces. $K_0$ and $K_1$ are rather classical objects from the 1950s; higher $K$-groups are defined by Quillen in two steps: to a Quillen exact category $C$ one first associates a $K$-theory space $𝒦(C)$ (or in better versions, a $K$-theory spectrum) and then defines $K$-groups as homotopy groups of that space:

$$K_i(C)={\pi }_i(𝒦(C)).$$K_i(C)=\pi_i(\mathcal{K}(C)).

The $K$-theory space of $C$ in Quillen’s version was obtained as a classifying space of the Quillen $Q$-construction applied to $C$. The $Q$-construction has been refined to more sophisticated delooping methods by Waldhausen, Karoubi and others.

References

• Charles Weibel, The K-Book: An introduction to algebraic K-theory (web)