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A spectral sequence for motivic cohomology, preprint (1995). , Stable real cohomology of arithmetic groups. Ann. Sci. Ecole Norm. Sup. (4) 7 (1974), 235–272. , Le th´eor`em de Riemann-Roch, Bull. Soc. Math. France 86 (1958) 97-136. , Surjectivity of cycle maps, in Journ´ ees de Geom´ etrie Alg´ ebrique D’Orsay, Juillet 1992, Ast´erisque 218 (1993) 203226. , Intersection theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 2. Springer-Verlag, Berlin-New York, 1984. , The Bloch-Kato conjecture and a theorem of SuslinVoevodsky, J.

2). 1) breaking up into short exact sequences jp∗ p/p+1 0 → Kq (MpX ) − → Kq (MX ∂p ) −→ Kq−1 (Mp+1 X ) → 0 for all p and q. 3. Cohomology of K-sheaves. One can go even further with this if one considers the K-sheaves Kp on X, defined as the sheaf associated to the presheaf U → Kp (U ). The stalk Kp,x at x ∈ X is just Kp (OX,x ). 3) 0 → Kq → ⊕x∈X (0) ix∗ (Kq (k(x))) − → d d 1 1 → ⊕x∈X (q) ix∗ (K0 (k(x))). ⊕x∈X (1) ix∗ (Kq−1 (k(x))) − → ... − Here ix : x → X is the inclusion, and we consider Kn (k(x)) as the constant sheaf on the one-point space x.

Conversely, let F : QE → Sets be a morphism-inverting functor with F (0) = F (M ) and F (iM ! ) = idM for all M . Given i : M M , we have i ◦ iM = iM , so F (i) = id. Suppose we have an exact sequence i j 0→M → − M→ − M → 0. ! ! , so ). Also jM = j ! ◦ jM , so F (j ! ) = F (jM Then j ! ◦ iM ! = i! jM ! ! F (jM ) = F (j ! )F (jM ) = F (jM )F (jM ). By the universal property of K0 , there is a unique group homomorphism ! ). This gives the inverse to of K0 (E) to Aut(F (0)) with [M ] → F (jM the transformation constructed above, proving the result.

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A short course in K-theory by Marc Levine

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