Basic theorems in algebraic K-theory
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
Additivity theorem[1] — Let [math]\displaystyle{ B, C }[/math] be exact categories (or other variants). Given a short exact sequence of functors [math]\displaystyle{ F' \rightarrowtail F \twoheadrightarrow F'' }[/math] from [math]\displaystyle{ B }[/math] to [math]\displaystyle{ C }[/math], [math]\displaystyle{ F_* \simeq F'_* + F''_* }[/math] as [math]\displaystyle{ H }[/math]-space maps; consequently, [math]\displaystyle{ F_* = F'_* + F''_*: K_i(B) \to K_i(C) }[/math].
The localization theorem generalizes the localization theorem for abelian categories.
Waldhausen Localization Theorem[2] — Let [math]\displaystyle{ A }[/math] be the category with cofibrations, equipped with two categories of weak equivalences, [math]\displaystyle{ v(A) \subset w(A) }[/math], such that [math]\displaystyle{ (A, v) }[/math] and [math]\displaystyle{ (A, w) }[/math] are both Waldhausen categories. Assume [math]\displaystyle{ (A, w) }[/math] has a cylinder functor satisfying the Cylinder Axiom, and that [math]\displaystyle{ w(A) }[/math] satisfies the Saturation and Extension Axioms. Then
- [math]\displaystyle{ K(A^w) \to K(A, v) \to K(A, w) }[/math]
is a homotopy fibration.
Resolution theorem[3] — Let [math]\displaystyle{ 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]\displaystyle{ K_i(C) = K_i(D) }[/math] for all [math]\displaystyle{ i \ge 0 }[/math].
Let [math]\displaystyle{ 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]\displaystyle{ 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.
Cofinality theorem[4] — Let [math]\displaystyle{ (A, v) }[/math] be a Waldhausen category that has a cylinder functor satisfying the Cylinder Axiom. Suppose there is a surjective homomorphism [math]\displaystyle{ \pi: K_0(A) \to G }[/math] and let [math]\displaystyle{ B }[/math] denote the full Waldhausen subcategory of all [math]\displaystyle{ X }[/math] in [math]\displaystyle{ A }[/math] with [math]\displaystyle{ \pi[X] = 0 }[/math] in [math]\displaystyle{ G }[/math]. Then [math]\displaystyle{ v.s. B \to v.s. A \to BG }[/math] and its delooping [math]\displaystyle{ K(B) \to K(A) \to G }[/math] are homotopy fibrations.
See also
References
- ↑ Weibel 2013, Ch. V, Additivity Theorem 1.2.
- ↑ Weibel 2013, Ch. V, Waldhausen Localization Theorem 2.1.
- ↑ Weibel 2013, Ch. V, Resolution Theorem 3.1.
- ↑ Weibel 2013, Ch. V, Cofinality Theorem 2.3.
- Weibel, Charles (2013). "The K-book: An introduction to algebraic K-theory". Graduate Studies in Math 145. http://www.math.rutgers.edu/~weibel/Kbook.html.
- 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
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