Homotopy invariance of stabilised algebraic K-theory

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

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

for a ring A need not induce an isomorphism on K₀ although it is a homotopy equivalence. Since ev0 is a split-surjection, the induced map K₀(A[t]) → K₀(A) is always surjective. Its kernel is denoted NK₀(A) (see [109, Definition 3.2.14]) and may be non-trivial. An example for this is A = ℂ[t²,t³] (see [109, Exercise 3.2.24]).