Basic theorems in algebraic K-theory

Four mathematical theorems

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

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Additivity theorem[1]—Let  B , C {\displaystyle B,C} be exact categories (or other variants). Given a short exact sequence of functors  F ′ ↣ F ↠ F ″ {\displaystyle F'\rightarrowtail F\twoheadrightarrow F''} from  B {\displaystyle B} to  C {\displaystyle C} ,  F ∗ ≃ F ∗ ′ + F ∗ ″ {\displaystyle F_{*}\simeq F'_{*}+F''_{*}} as  H {\displaystyle H} -space maps; consequently,  F ∗ = F ∗ ′ + F ∗ ″ : K i ( B ) -> K i ( C ) {\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  A {\displaystyle A} be the category with cofibrations, equipped with two categories of weak equivalences,  v ( A ) ⊂ w ( A ) {\displaystyle v(A)\subset w(A)} , such that  ( A , v ) {\displaystyle (A,v)} and  ( A , w ) {\displaystyle (A,w)} are both Waldhausen categories. Assume  ( A , w ) {\displaystyle (A,w)} has a cylinder functor satisfying the Cylinder Axiom, and that  w ( A ) {\displaystyle w(A)} satisfies the Saturation and Extension Axioms. Then

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

is a homotopy fibration.

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

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

## See also

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* Fundamental theorem of algebraic K-theory

## References

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1. ^ Weibel 2013, Ch. V, Additivity Theorem 1.2.
2. ^ Weibel 2013, Ch. V, Waldhausen Localization Theorem 2.1.
3. ^ Weibel 2013, Ch. V, Resolution Theorem 3.1.
4. ^ Weibel 2013, Ch. V, Cofinality Theorem 2.3.

## Bibliography

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* Weibel, Charles (2013). "The K-book: An introduction to algebraic K-theory". Graduate Studies in Math. Graduate Studies in Mathematics. 145. doi:10.1090/gsm/145. ISBN 978-0-8218-9132-2.
* Ross E. Staffeldt, On Fundamental Theorems of Algebraic K-Theory
* GABE ANGELINI-KNOLL, FUNDAMENTAL THEOREMS OF ALGEBRAIC K-THEORY
* Harris, Tom (2013). "Algebraic proofs of some fundamental theorems in algebraic K-theory". arXiv:1311.5162 [math.KT].

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