Basic theorems in algebraic K-theory explained

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

The localization theorem generalizes the localization theorem for abelian categories.

Let

C \subset D 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 M ⊕ N is in C. The prototypical example is when C is the category of free modules and D is the category of projective modules.

See also

• Fundamental theorem of algebraic K-theory

Bibliography

• Charles . Weibel . The K-book: An introduction to algebraic K-theory . Graduate Studies in Math . Graduate Studies in Mathematics . 145 . 2013. 10.1090/gsm/145 . 978-0-8218-9132-2 .
• Ross E. Staffeldt, On Fundamental Theorems of Algebraic K-Theory
• GABE ANGELINI-KNOLL, FUNDAMENTAL THEOREMS OF ALGEBRAIC K-THEORY
• 1311.5162 . Harris . Tom . Algebraic proofs of some fundamental theorems in algebraic K-theory . 2013 . math.KT .