Basic theorems in algebraic K-theory: Difference between revisions

Revision as of 05:58, 29 October 2019

Revision as of 05:59, 29 October 2019

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The proof uses [[Quillen's Theorem A]]?-->

Let <math>C \subset D</math> be exact categories. Then ''C'' is said to be '''[[cofinal subcategory|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 <math>M \oplus N</math> is in ''C''. The prototypical example is when ''C'' is the category of [[free modulesmodule]]s and ''D'' is the category of [[projective modulesmodule]]s.

{{math_theorem|name=Cofinality theorem<ref>{{harvnb|Weibel|loc=Ch. V, Cofinality Theorem 2.3.}}</ref>|Let <math>(A, v)</math> be a Waldhausen category that has a cylinder functor satisfying the Cylinder Axiom. Suppose there is a surjective homomorphism <math>\pi: K_0(A) \to G</math> and let <math>B</math> denote the full Waldhausen subcategory of all <math>X</math> in <math>A</math> with <math>\pi[X] = 0</math> in <math>G</math>. Then <math>v.s. B \to v.s. A \to BG</math> and its delooping <math>K(B) \to K(A) \to G</math> are homotopy fibrations.}}