Basic theorems in algebraic K-theory

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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 ${\displaystyle B,C}$  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 an abelian categories.

Localization theorem—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—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—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.

References

• 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