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We now take the case X = BGL(R), N = E(R), forming i : BGL(R) → BGL(R)+ . By our identification of H∗ (GL(R), Z) with H∗ (BGL(R), Z), we thus have the canonical identification H∗ (BGL(R)+ , Z) ∼ = H∗ (GL(R), Z). 3. Let R be a ring. The higher K-groups Ki (R), i = 1, 2, . . are defined by Ki (R) := πi (BGL(R)+ , ∗). 3. K1 and K2 . We need to reconcile this definition with the “classical” definition of K1 and K2 . 4. For i = 1, 2, the new definition of Ki (R) agrees with the old one. Proof. 1)(1).

The K∗ (X)-ring structure is natural with respect to the pull-back maps f ∗ , and, for f : Y → X projective, we have the projection formula: f∗ (f ∗ (a) · b) = a · f∗ (b); a ∈ Ki (X), b ∈ Gj (Y ). 5. Projective bundles. Both K-theory and G-theory satisfy a projective bundle formula: for E → X a vector bundle of rank r + 1, K∗ (P(E)) is a free K∗ (X)-module with basis 1, [O(−1)], . . , [O(−r)], and similarly for G-theory; the proof is essentially the same as for G0 and K0 . There is an interesting extension of this to “twisted forms” of projective bundles, the so-called Severi-Brauer varieties over K.

341, Springer, Berlin, 1973. , Rational equivalence of zero-cycles, Math. USSR Sbornik 18 (1972), 571-588. , The torsion of the group of 0-cycles modulo rational equivalence, Ann. of Math. 111 (1979) 415-569. [34] Rubin, K. The “main conjectures” of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1991), no. 1, 25–68. , Alg` ebre locale. Multiplicit´ es. Cours au Coll`ege de France, 1957–1958, r´edig´e par Pierre Gabriel. Lecture Notes in Mathematics, 11 Springer-Verlag, Berlin-New York 1965 [36] Soul´e, Christophe.

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