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 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

  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.
  • Weibel, Charles (2013). "The K-book: An introduction to algebraic K-theory". Graduate Studies in Math. 145.
  • 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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