Homotopy invariance of stabilised algebraic K-theory

Part of the Oberwolfach Seminars book series (OWS, volume 36)


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]).


Exact Sequence Ring Endomorphism Stable Functor Homotopy Invariance Split Extension 
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© Birkhäuser Verlag AG 2007

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