A short course in K-theory by Marc Levine

By Marc Levine

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N, σin (j) = j j−1 if j < i if j ≥ i. These satisfy certain relations, which we don’t specify here. A simplicial object S is thus often given by defining the n-simplices Sn , the boundary maps ∂in = S(δin−1 ) : Sn → Sn−1 , and the degeneracy maps sni = S(σin+1 ) : Sn → Sn+1 . The fundamental example of a cosimplicial space (C = Top) is ∆ : Ord → Top. ∆([n]) is the standard topological n-simplex: n ∆([n]) = ∆n := {(t0 , . . , tn ) ∈ R n+1 | ti = 1, ti ≥ 0}. i=0 ∆n has vertices v0n , . . , vnn , where vin has ti = 1, tj = 0 for j = i; clearly ∆n is the convex hull of its vertices.

P 1 → P0 → M → 0 with the Pi in E0 . Then BQi : BQ(E0 ) → BQ(E1 ) induces an isomorphism i∗ : Kp (E0 ) → Kp (E1 ) for all p. 9 ([30, Theorem 5, §5]). Let i : B → A be the inclusion of a Serre subcategory B of an abelian category A, and let j : A → A/B be the canonical quotient map. Then BQi BQj BQ(B) −−→ BQ(A) −−→ BQ(A/B) is a weak homotopy fiber sequence, so we have a long exact sequence of K-groups ∂ j∗ i ∗ . . → Kp+1 (A/B) → − Kp (B) − → Kp (A) − → Kp (A/B) → . . In addition K0 (A) → K0 (A/B) is surjective.

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. , On the anticyclotomic main conjecture for CM fields, Invent.

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