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[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, ${\displaystyle F_{*}=F'_{*}+F''_{*}:K_{i}(B)\to K_{i}(C)}$.

The localization theorem generalizes the localization theorem for abelian categories.

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 ${\displaystyle i\geq 0}$.

Let ${\displaystyle C\subset D}$ 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 ${\displaystyle M\oplus N}$ 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[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.

See also

• Fundamental theorem of algebraic K-theory

References

1. Weibel, Ch. V, Additivity Theorem 1.2. harvnb error: no target: CITEREFWeibel (help)
2. Weibel, Ch. V, Waldhausen Localization Theorem 2.1. harvnb error: no target: CITEREFWeibel (help)
3. Weibel, Ch. V, Resolution Theorem 3.1. harvnb error: no target: CITEREFWeibel (help)
4. Weibel, Ch. V, Cofinality Theorem 2.3. harvnb error: no target: CITEREFWeibel (help)

• C. Weibel "The K-book: An introduction to algebraic K-theory"
• Ross E. Staffeldt, On Fundamental Theorems of Algebraic K-Theory
• GABE ANGELINI-KNOLL, FUNDAMENTAL THEOREMS OF ALGEBRAIC K-THEORY
• Tom Harris, Algebraic proofs of some fundamental theorems in algebraic K-theory