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

By Daniel Dugger

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Proof. For the first claim simply observe that A 0 X A = B 0 A−1 X . 9(a), and hence in [GL(R), GL(R)] by the Whitehead Lemma. 9(b) together with the Whitehead Lemma. 8. Let α : P → P be an automorphism of a finitely-generated projective, and let Q be a free complement to P : that is, P ⊕ Q ∼ = Rn for some n. Then α⊕idQ α [P −→ P ] = [P ⊕ Q −→ P ⊕ Q] in K1sp (R), which shows that K1sp,f r (R) → K1sp (R) is surjective. 7). The fact that colimn GLn (R)ab → colimP Aut(P )ab is an isomorphism is very easy: it is just because the subcategory of T (M) consisting of the free modules is cofinal in T (M).

E(R). (c) Let A be a matrix obtained from the identity by switching two colums and multiplying one of the switched columns by −1. Then A ∈ E(R), and similarly for the transpose of A. Proof. For part (a) just note that I0 X can be obtained from the identity matrix I by a sequence of allowable column operations of the type discussed above. For the transpose, use row operations. For (b) consider the following chain of matrices: I 0 0 I ∼ I 0 A I ∼ I A−2 − A−1 A 0 ∼ 0 A−1 A A ∼ A−2 − A−1 A−1 0 . A−1 Passage from each matrix to the next can be done by allowable row and column operations; alternatively, each matrix can be obtained from its predecessor by left or right multiplication by a matrix of the type considered in (a).

It should be thought of as listing all the zeros of the “function” r, together with their orders of vanishing (see below for an example). For a general element x ∈ F ∗ we would just write x = r/s for r, s ∈ D − {0}, and then ∂(x) = div(r) − div(s); this gives the zeros and poles of x, with multiplicities. The quotient of P =0 Z by the classes div(x) is called the divisor class group of D; it is isomorphic to the ideal class group from algebraic number theory. Our short exact sequence shows that K0 (D) is also isomorphic to this group.

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