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 $F^{\prime }\rightarrowtail F\twoheadrightarrow F^{″}$ from $B$ to $C$, $F_{*}\simeq F{\text{'}}_{*}+F^{\prime }{\text{'}}_{*}$ as $H$-space maps; consequently, $F_{*}=F{\text{'}}_{*}+F^{\prime }{\text{'}}_{*}:K_i(B)\to K_i(C)$.

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 $i\ge 0$.

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.

See also

• Fundamental theorem of algebraic K-theory

Bibliography

• Ross E. Staffeldt, On Fundamental Theorems of Algebraic K-Theory
• GABE ANGELINI-KNOLL, FUNDAMENTAL THEOREMS OF ALGEBRAIC K-THEORY