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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. Reine Angew. Math. 530 (2001), 55–103. , La th´eorie de classes de Chern, Bull. Soc. Math. France 86 (1958) 137-154.

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. 3) is exact. 3) gives a flasque resolution of the sheaf Kq . 3. Suppose Gersten’s conjecture is true for all the local rings OX,x . 2). In particular, H p (X, Kq ) = 0 if p > q. 3) (after deleting Kq ). 4. Take the case q = 1. Then K1 is just the sheaf of units × OX , and the sheafified Gersten complex is (assume X is irreducible with generic point η) × 1 → OX → iη∗ k(X)× → ⊕x∈X (1) ix∗ Z → 0.

As one consequence, let X be a scheme, and take Z = Xred . Then MX (Z) = MX , and thus Gp (Xred ) → Gp (X) is an isomorphism. 3. , X is a smooth scheme over a field). Then the inclusion PX → MX induces an isomorphism Kp (X) → Gp (X). 46 A SHORT COURSE IN K-THEORY MEXICO CITY MAY, 2002 Indeed, if X is regular and noetherian, each coherent sheaf F on X has a finite resolution by locally free sheaves. 8. 4. Let i : Z → X be the inclusion of a closed subscheme, j : U → X the open complement. Then the sequence BQj ∗ BQi∗ BQ(MZ ) −−−→ BQ(MX ) −−−→ BQ(MU ) is a weak homotopy fiber sequence, so we have a long exact sequence ∂ i j∗ ∗ → Gp (U ) → .

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A short course in K-theory by Marc Levine


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