Basic theorems in algebraic K-theory

Short description: 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).

## Contents

* 1 Theorems
* 2 See also
* 3 References
* 4 Bibliography

## Theorems

Additivity theorem[1] -- Let B,C be exact categories (or other variants). Given a short exact sequence of functors F′↣F↠F″ from B to C, F*≃F'*+F′'* as H-space maps; consequently, F*=F'*+F′'*:Ki(B)->Ki(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)⊂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(Aw)->K(A,v)->K(A,w)

is a homotopy fibration.

Resolution theorem[3] -- Let C⊂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 Ki(C)=Ki(D) for all i≥0.

Let C⊂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 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 π:K0(A)->G and let B denote the full Waldhausen subcategory of all X in A with π[X]=0 in G. Then v.s.B->v.s.A->BG and its delooping K(B)->K(A)->G are homotopy fibrations.

## See also

* Fundamental theorem of algebraic K-theory

## References

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

* 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. http://www.math.rutgers.edu/~weibel/Kbook.html.
* 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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