By Daniel Dugger

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

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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. Just as in the previous example, the image of K1 (D) in F ∗ is just D∗ . The map K0 (D) → Z is just the usual rank map, so its kernel is K0 (D). So we get a short exact sequence ∂ 0 → F ∗ /D∗ −→ K0 (D, S) → K0 (D) → 0. 48 DANIEL DUGGER We know K0 (D, S) ∼ = G(M | S −1 M = 0). The condition S −1 M = 0 just says that M is a torsion module.

If P• is a bounded, exact complex of projectives then it gives rise to a relation in K(R), and (equivalently) represents the zero object in K cplx (R). Given this, it might seem surprising to learn that there is yet another model for K(R) in which exact complexes can represent nonzero elements—and even more, all nonzero elements can be represented this way. The goal of the present section is to explain this model, as well as some variations. This material is adapted from [Gr]. Note: The contents of this section are only needed once in the remainder of the notes, for a certain perspective on Adams operations in Section 30.

Take u to be the product of all the denominators of the entries in A). Then uA represents a map β : Rn → Rn , and we have the commutative diagram (S −1 R)n y β G (S −1 R)n G (S −1 R)n y uIn G Rn β Rn where the vertical maps are localization. This diagram gives uIn ◦ β ∼ = S −1 β , and −1 −1 so [uIn ] + [β] = [S β ] in K1 (S R). 16) [β] = [S −1 β ] − n[S −1 u]. This shows that K1 (R) is generated by classes [S −1 α] for α : Rn → Rn , and we have thereby proven the uniqueness part of the proposition.

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

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