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
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Additivity theorem[1] — Let be exact categories (or other variants). Given a short exact sequence of functors from to , as -space maps; consequently, .
The localization theorem generalizes the localization theorem for abelian categories.
Waldhausen Localization Theorem[2] — Let be the category with cofibrations, equipped with two categories of weak equivalences, , such that and are both Waldhausen categories. Assume has a cylinder functor satisfying the Cylinder Axiom, and that satisfies the Saturation and Extension Axioms. Then
is a homotopy fibration.
Resolution theorem[3] — Let 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 for all .
Let 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 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 be a Waldhausen category that has a cylinder functor satisfying the Cylinder Axiom. Suppose there is a surjective homomorphism and let denote the full Waldhausen subcategory of all in with in . Then and its delooping are homotopy fibrations.
See also
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