Basic theorems in 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.)

Additivity theorem—Let

Localization theorem—Let

Resolution theorem—Let ${\displaystyle C\subset D}$ 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 ${\displaystyle K_{i}(C)=K_{i}(D)}$ for all ${\displaystyle i\geq 0}$.

Let ${\displaystyle C\subset D}$ 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 ${\displaystyle M\oplus N}$ is in C. The prototypical example is when C is the category of free modules and D is the category of projective modules.

Cofinality theorem—Let

References

• C. Weibel "The K-book: An introduction to algebraic K-theory"
• Ross E. Staffeldt, On Fundamental Theorems of Algebraic K-Theory
• GABE ANGELINI-KNOLL, FUNDAMENTAL THEOREMS OF ALGEBRAIC K-THEORY
• Tom Harris, Algebraic proofs of some fundamental theorems in algebraic K-theory