Download PDF by Marc Levine: A short course in K-theory

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2. The Quillen spectral sequence. Gersten’s conjecture can be viewed as a kind of local triviality of the K-theory functor for regular schemes, anaologous to the assertion that the singular cohomology of an open disk is trivial. 1) jp∗ p/p+1 p . . → Kq (Mp+1 → Kq (MX X ) → Kq (MX ) − ∂p ) −→ Kq−1 (Mp+1 X ) → ... with the similar one coming from the sequence Mp+2 → Mp+1 → X X p+1/p+2 p/p+1 p+1/p+2 p,−p−q MX to define the map d1 : Kq (MX ) → Kq−1 (MX ) as the composition p/p+1 Kq (MX ∗ jp+1 ∂p p+1/p+2 ) −→ Kq−1 (Mp+1 X ) −−→ Kq−1 (MX ).

Math. France 86 (1958) 137-154. , On the anticyclotomic main conjecture for CM fields, Invent. Math. 117 (1994), no. 1, 89–147. , Higher algebraic K-theory. II (after Daniel Quillen). Algebraic K-theory (Proc. , 1976), pp. 217–240. , 551, Springer, Berlin, 1976. [16] Hiller, Howard L. λ-rings and algebraic K-theory. J. Pure Appl. Algebra 20 (1981), no. 3, 241–266. , Mixed motives, Lecture Notes in Math. 1400, Springer, Berlin 1990. , Zero-cycles and complete intersections on affine surfaces, J.

Form X + by attaching 3-cells Dα3 to Y by the attaching maps p∗ fα . ˜ + similarly by attaching D 3 with attaching maps g ·fα , g ∈ G. Form X α,g ˜ + → X + is again covering space with group G. Then X 34 A SHORT COURSE IN K-THEORY MEXICO CITY MAY, 2002 Since π1 (X + ) = π1 (Y ), we have π1 (X + ) = G. Now let L be a local system on X + , and let L be the corresponding G-module. As the ˜ + , X) ˜ is clearly relative chain complex C∗ (X ∼ = 0 → ... → F − → F → . . → 0, the relative chain complex C∗ (X + , X; L) is thus ∼ = 0 → .

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A short course in K-theory by Marc Levine

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