By Daniel Dugger

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**Extra resources for A geometric introduction to K-theory [Lecture notes]**

**Example text**

It suffices to prove that F (β, tu) = F (β, u) for any t ∈ S; for if u is another choice for u then 46 DANIEL DUGGER we would have F (β, u) = F (β, u u) = F (β, u ). 12, applied twice). Let us now write F (β) instead of F (β, u). The last thing that must be checked is that F (β ⊕ β ) = F (β) + F (β ), but this is obvious. So we have established the existence of ∂ : K1 (R) → K0 (R, S) having the desired properties. 17 (Localization sequence for K-theory). Let R be a commutative ring and S ⊆ R a multiplicative system.

A GEOMETRIC INTRODUCTION TO K-THEORY 39 If M is an n-multicomplex then let CM denote the cone on the identity map M → M . This is an (n + 1)-multicomplex, defined in the evident manner. This cone construction induces a group homomorphism K n−exct (R) → K (n+1)−exct (R). 10. The map K n−exct (R) → K (n+1)−exct (R) is an isomorphism, with inverse given by χ (M ) = (−1)j+1 j[Mj, ] where the symbols Mj, represent the various slices of M in any fixed direction. Proof. 2 almost verbatim, but where each Pi represents an n-exact multicomplex rather than an R-module.

This cone construction induces a group homomorphism K n−exct (R) → K (n+1)−exct (R). 10. The map K n−exct (R) → K (n+1)−exct (R) is an isomorphism, with inverse given by χ (M ) = (−1)j+1 j[Mj, ] where the symbols Mj, represent the various slices of M in any fixed direction. Proof. 2 almost verbatim, but where each Pi represents an n-exact multicomplex rather than an R-module. We have the sequence of isomorphisms K(R) → K exct (R) → K 2−exct (R) → · · · The composite map K(R) → K n−exct (R) sends [P ] to the n-dimensional cube consisting of P ’s and identity maps.

### A geometric introduction to K-theory [Lecture notes] by Daniel Dugger

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