April 5, 2017

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By Marc Levine

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5. Take g ∈ GL(R). Then the conjugation action by g on BGL(R) extends to an action on BGL(R)+ which is homotopic to the identity. Additionally, one can show that the stable sum ⊕σ extends to an operation ⊕ : BGL(R)+ ×BGL(R)+ → BGL(R)+ which makes BGL(R)+ into an H-group, with the identity matrix as unit. Loday [21] has shown that the tensor product operation on matrices (A, B) → A ⊗ B can be modified to define a product BGL(R)+ ∧ BGL(R)+ → BGL(R)+ which makes the graded group ⊕∞ i=0 Ki (R) into a graded ring.

Gp+1 (U ) → − Gp (Z) − → Gp (X) − for p ≥ 0. Also, j ∗ : G0 (X) → G0 (U ) is surjective. Proof. We have the equivalence of categories MU ∼ MX /MX (Z). 9. 2, BQ(MZ ) → BQ(MX (Z)) is a weak equivalence, which proves the theorem. 5. We now have two possible definitions of the K-theory of a commutative ring R, namely Kp (R) and Kp (Spec R). In [15], it is shown that there is a natural isomorphism of these two, even as H-spaces. 2. Mayer-Vietoris. 6. Let X be a scheme, jU : U → X, jV : V → X open subschemes with X = U ∪ V .

The results mentioned in the previous section have inspired a number of conjectures, some of which remain open to this day. 9 (Bass). Let R be a commutative ring which is finitely generated over Z. Then K0 (R) is a finitely generated group. The Bass conjecture for R the ring of S-integers in a number field follows from the finitenes of the class group. For R the ring of a curve over a finite field, the conjecture follows from the representability of the Picard group of a smooth projective curve by the Jacobian times Z.

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