Topological and Bivariant K-Theory pp 45-62 | Cite as

# Homotopy invariance of stabilised algebraic K-theory

Chapter

## Abstract

There are many interesting algebras that are not local Banach algebras (see Exercise 2.14), so that the results of Chapter 2 do not apply to them. Problems with homotopy invariance already occur in a purely algebraic context: the evaluation homomorphism
for a ring

$$
ev_0 :A[t]: = A \otimes _\mathbb{Z} \mathbb{Z}[t] \to A
$$

*A*need not induce an isomorphism on K_{0}although it is a homotopy equivalence. Since ev0 is a split-surjection, the induced map K_{0}(*A*[*t*]) → K_{0}(*A*) is always surjective. Its kernel is denoted NK_{0}(*A*) (see [109, Definition 3.2.14]) and may be non-trivial. An example for this is*A*= ℂ[*t*^{2},*t*^{3}] (see [109, Exercise 3.2.24]).## Keywords

Exact Sequence Ring Endomorphism Stable Functor Homotopy Invariance Split Extension
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Birkhäuser Verlag AG 2007