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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

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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

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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].