# Daniel Dugger's A geometric introduction to K-theory [Lecture notes] PDF By Daniel Dugger

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Example text

Define χ : G(M | S −1 M = 0) → G0 (R/P ) P =0 by sending [M ] to the tuple of integers DP (MP ), as P runs over all maximal ideals of D (the only ones that give nonzero lengths are the ones containing Ann M , and there are only finitely-many of these since they are precisely the minimal primes of Ann M ). It is easy to check that χ ◦ j = id; since j was already known to be surjective this means they are inverse isomorphisms. So we can rewrite our short exact sequence as ∂ 0 → F ∗ /D∗ −→ Z → K0 (D) → 0.

Note that (R/πR) = 1. We can write r = uπ k for some unit u ∈ R and n ≥ 0, in which case R/rR ∼ = R/π k R and so (R/rR) = k. It follows that the composite ∂ ∼ = ∼ = F ∗ −→ K0 (R, S) −→ G(M | S −1 M = 0) −→ Z is just the usual π-adic valuation on F ∗ . The following example generalizes the previous one, but is a bit more interesting. 19. Let D be a Dedekind domain—a regular ring of dimension one. In such a ring all nonzero primes are maximal ideals. Let S = D − {0} and let F = S −1 D be the quotient field.

10. (a) If f is an inclusion Y → X then f ∗ E is just the restriction E|Y that we have discussed before. (b) Pullback bundles can be slightly non-intuitive. Let M → S 1 be the M¨obius bundle, and let f : S 1 → S 1 be the map z → z 2 . We claim that f ∗ M ∼ = 1. This is easiest to see if one uses the following model for M : M= θ eiθ , rei 2 θ ∈ [0, 2π], r ∈ R . 54 DANIEL DUGGER The bundle map is projection onto the first coordinate π : M → S 1 . Then f ∗ M = {(eiθ , reiθ ) | θ ∈ [0, 2π], r ∈ R}.