Basic theorems in algebraic K-theory: Difference between revisions

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{{DISPLAYTITLE:Basic theorems in algebraic ''K''-theory}}

#REDIRECT [[Algebraic K-theory]]

{{short description|Four mathematical theorems}}

In mathematics, there are several theorems basic to [[algebraic K-theory|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 subcategory|isomorphism-closed]]).

== Theorems ==

{{expand section|date=October 2019}}<!-- need much more explanations. -->

{{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''.

Then <math>K_i(C) = K_i(D)</math> for all <math>i \ge 0</math>.}}

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By induction, the proof reduces to the case when

*(i) objects in ''C''' have length one resolution by objects in ''C'' or

*(ii) there is a filtration so that ([[dévissage]])}}

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 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 ==

*[[Fundamental theorem of algebraic K-theory]]

== References ==

{{reflist}}

== Bibliography ==

*{{cite journal |first=Charles |last=Weibel |author-link=Charles 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 |series=Graduate Studies in Mathematics |volume=145 |year=2013|doi=10.1090/gsm/145 |isbn=978-0-8218-9132-2 |url-access=subscription }}

*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]

*{{cite arXiv|eprint=1311.5162 |last1=Harris |first1=Tom |title=Algebraic proofs of some fundamental theorems in algebraic ''K''-theory |date=2013 |class=math.KT }}

#REDIRECT [[Category:Algebraic K-theory]]

[[Category:Theorems in algebraic topology]]

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