Basic theorems in algebraic K-theory

Additivity theorem[1] — Let ${\displaystyle B,C}$ be exact categories (or other variants). Given a short exact sequence of functors ${\displaystyle F'\rightarrowtail F\twoheadrightarrow F''}$ from ${\displaystyle B}$ to ${\displaystyle C}$, ${\displaystyle F_{*}\simeq F'_{*}+F''_{*}}$ as ${\displaystyle H}$-space maps; consequently, ${\displaystyle F_{*}=F'_{*}+F''_{*}:K_{i}(B)\to K_{i}(C)}$.

Waldhausen Localization Theorem[2] — Let ${\displaystyle A}$ be the category with cofibrations, equipped with two categories of weak equivalences, ${\displaystyle v(A)\subset w(A)}$, such that ${\displaystyle (A,v)}$ and ${\displaystyle (A,w)}$ are both Waldhausen categories. Assume ${\displaystyle (A,w)}$ has a cylinder functor satisfying the Cylinder Axiom, and that ${\displaystyle w(A)}$ satisfies the Saturation and Extension Axioms. Then

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

is a homotopy fibration.

Resolution theorem[3] — Let ${\displaystyle C\subset D}$ 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 ${\displaystyle K_{i}(C)=K_{i}(D)}$ for all ${\displaystyle i\geq 0}$.

Cofinality theorem[4] — Let ${\displaystyle (A,v)}$ be a Waldhausen category that has a cylinder functor satisfying the Cylinder Axiom. Suppose there is a surjective homomorphism ${\displaystyle \pi :K_{0}(A)\to G}$ and let ${\displaystyle B}$ denote the full Waldhausen subcategory of all ${\displaystyle X}$ in ${\displaystyle A}$ with ${\displaystyle \pi [X]=0}$ in ${\displaystyle G}$. Then ${\displaystyle v.s.B\to v.s.A\to BG}$ and its delooping ${\displaystyle K(B)\to K(A)\to G}$ are homotopy fibrations.