Basic theorems in algebraic K-theory
# Basic theorems in algebraic K-theory

In mathematics, there are several theorems basic to algebraic K-theory.

Throughout, for simplicity, we assume when an exact category is a subcategory of another exact category, we mean it is strictly full subcategory (i.e., isomorphism-closed.)

## Theorems

Additivity theorem — Let $B,C$ be exact categories (or other variants). Given a short exact sequence of functors $F'\rightarrowtail F\twoheadrightarrow F''$ from $B$ to $C$, $F_{*}\simeq F'_{*}+F''_{*}$ as $H$-space maps; consequently, $F_{*}=F'_{*}+F''_{*}:K_{i}(B)\to K_{i}(C)$.

The localization theorem generalizes the localization theorem for abelian categories.

Waldhausen Localization Theorem — Let $A$ be the category with cofibrations, equipped with two categories of weak equivalences, $v(A)\subset w(A)$, such that $(A,v)$ and $(A,w)$ are both Waldhausen categories. Assume $(A,w)$ has a cylinder functor satisfying the Cylinder Axiom, and that $w(A)$ satisfies the Saturation and Extension Axioms. Then

    $K(A^{w})\to K(A,v)\to K(A,w)$

is a homotopy fibration.

Resolution theorem — Let $C\subset D$ be exact categories. Assume

* (i) C is closed under extensions in D and under the kernels of admissible surjections in D.
* (ii) Every object in D admits a resolution of finite length by objects in C.

Then $K_{i}(C)=K_{i}(D)$ for all $i\geq 0$.

Let $C\subset D$ be exact categories. Then C is said to be cofinal in D if (i) it is closed under extension in D and if (ii) for each object M in D there is an N in D such that $M\oplus N$ is in C. The prototypical example is when C is the category of free modules and D is the category of projective modules.

Cofinality theorem — Let $(A,v)$ be a Waldhausen category that has a cylinder functor satisfying the Cylinder Axiom. Suppose there is a surjective homomorphism $\pi :K_{0}(A)\to G$ and let $B$ denote the full Waldhausen subcategory of all $X$ in $A$ with $\pi [X]=0$ in $G$. Then $v.s.B\to v.s.A\to BG$ and its delooping $K(B)\to K(A)\to G$ are homotopy fibrations.

