Basic theorems in algebraic K-theory: Difference between revisions

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#REDIRECT [[Algebraic K-theory]]

'''Basic theorems in 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.)

{{math_theorem|name=Additivity theorem|Let}}

{{math_theorem|name=Localization theorem|Let}}

{{math_theorem|name=Resolution theorem|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>.}}

<!--

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''' 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 modules and ''D'' is the category of projective modules.

{{math_theorem|name=Cofinality theorem|Let}}

== References ==

*C. Weibel "[http://www.math.rutgers.edu/~weibel/Kbook.html The K-book: An introduction to algebraic K-theory]"

*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]

*Tom Harris, [https://arxiv.org/abs/1311.5162 Algebraic proofs of some fundamental theorems in algebraic K-theory]