Basic theorems in algebraic K-theory: Difference between revisions

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{{math_theorem|name=Additivity theorem<ref>{{harvnb|Weibel|2013|loc=Ch. V, Additivity Theorem 1.2.}}</ref>|Let <math>B, C</math> be exact categories (or other variants). Given a short exact sequence of functors <math>F' \rightarrowtail F \twoheadrightarrow F''</math> from <math>B</math> to <math>C</math>, <math>F_* \simeq F'_* + F''_*</math> as <math>H</math>-space maps; consequently, <math>F_* = F'_* + F''_*: K_i(B) \to K_i(C)</math>.}}

The localization theorem generalizes the [[localization theorem for abelian categories]].

{{math_theorem|name=Waldhausen Localization Theorem<ref>{{harvnb|Weibel|2013|loc=Ch. V, Waldhausen Localization Theorem 2.1.}}</ref>|Let <math>A</math> be the category with cofibrations, equipped with two categories of weak equivalences, <math>v(A) \subset w(A)</math>, such that <math>(A, v)</math> and <math>(A, w)</math> are both Waldhausen categories. Assume <math>(A, w)</math> has a [[cylinder functor]] satisfying the Cylinder Axiom, and that <math>w(A)</math> satisfies the Saturation and Extension Axioms. Then

:<math>K(A^w) \to K(A, v) \to K(A, w)</math>

is a [[homotopy fibration]].

}}

{{math_theorem|name=Resolution theorem<ref>{{harvnb|Weibel|2013|loc=Ch. V, Resolution Theorem 3.1.}}</ref>|Let <math>C \subset D</math> 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''.

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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 module]]s and ''D'' is the category of [[projective module]]s.

{{math_theorem|name=Cofinality theorem<ref>{{harvnb|Weibel|2013|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.}}

== See also ==

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== References ==

{{reflist}}

*C.{{cite journal |first=Charles |last=Weibel "[|url=http://www.math.rutgers.edu/~weibel/Kbook.html |title=The K-book: An introduction to algebraic K-theory]" |journal=Graduate Studies in Math |volume=145 |year=2013}}

*Ross E. Staffeldt, [http://folk.uio.no/rognes/kurs/mat9570v10/S89.pdf On Fundamental Theorems of Algebraic K-Theory]

*GABE ANGELINI-KNOLL, [http://www.math.wayne.edu/~gak/talks/FTKthytalk.pdf FUNDAMENTAL THEOREMS OF ALGEBRAIC K-THEORY]