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 $B,C$ 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^[2] — 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^[3] — 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^[4] — 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.

References

^ Weibel, Ch. V, Additivity Theorem 1.2.
^ Weibel, Ch. V, Waldhausen Localization Theorem 2.1.
^ Weibel, Ch. V, Resolution Theorem 3.1.
^ Weibel, Ch. V, Cofinality Theorem 2.3.

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

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[1] Weibel, Ch. V, Additivity Theorem 1.2.

[2] Weibel, Ch. V, Waldhausen Localization Theorem 2.1.

[3] Weibel, Ch. V, Resolution Theorem 3.1.

[4] Weibel, Ch. V, Cofinality Theorem 2.3.

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Theorems

See also

References