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

By Daniel Dugger

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The map GL(R)ab → K1 (R) is an isomorphism. It will be convenient to prove this at the same time that we give other descriptions for K1 (R). In particular, we make the following definitions: (1) K1f r (R) is the group defined similarly to K1 (R) but changing all occurrences of ‘projective’ to ‘free’. 5). The “sp” stands for “split”. (3) K1sp,f r (R) is the group defined by making both the changes indicated in (1) and (2). 7) GL(R)ab colimP Aut(P )ab y G G K sp (R) 1 y G G K1 (R) y colimn GLn (R)ab G G K sp,f r (R) 1 G G K f r (R).

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. Our localization sequence looks like K1 (D) → F ∗ → K0 (D, S) → K0 (D) → Z.

Proof. Use the following short exact sequence of maps: 0 GP f 0 g idQ ⊕βα α  GQ G Q⊕P f  G Q⊕W g GQ G0 β  GW G 0, A GEOMETRIC INTRODUCTION TO K-THEORY 33 where f (x) = (α(x), x), f (y) = (y, β(y)), g(a, b) = a − α(b), and g (c, d) = β(c) − d. This gives that βα id β α [Q −→ Q] + [P −→ W ] = [P −→ Q] + [Q −→ W ], but of course the first term on the left is zero in K(R, S). Note that there is an evident map K(R, S) → K(R) that sends a class [P ] in K(R, S) to the similarly-named (but different) class [P ] in K(R); in colloquial terms, the map simply ‘forgets’ that a complex P is S-exact.

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