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 ⊂ D {\displaystyle 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 {\displaystyle M\oplus 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
References
Bibliography Weibel, Charles (2013). "The K-book: An introduction to algebraic K-theory". Graduate Studies in Math. Graduate Studies in Mathematics. 145. doi:10.1090/gsm/145. ISBN 978-0-8218-9132-2.{{cite journal}}: CS1 maint: periodical has ISBN (link) Ross E. Staffeldt, On Fundamental Theorems of Algebraic K-Theory GABE ANGELINI-KNOLL, FUNDAMENTAL THEOREMS OF ALGEBRAIC K-THEORY Harris, Tom (2013). "Algebraic proofs of some fundamental theorems in algebraic K-theory". arXiv:1311.5162 [math.KT].