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Algebraic cycles and vector bundles over affine varieties, Algebra and its applications (New Delhi, 1981), 25–28, Lecture Notes in Pure and Appl. , 91, Dekker, New York, 1984. [28] Mumford, D. Rational equivalence of zero-cycles on surfaces, J. Math. Kyoto Univ. 9 (1968), 195-204. , Zero cycles and projective modules, Ann. of Math. (2) 140 (1994), no. 2, 405–434. 56 A SHORT COURSE IN K-THEORY MEXICO CITY MAY, 2002 [30] Quillen, Daniel. Higher algebraic K-theory. I. Algebraic K-theory, I: Higher K-theories (Proc.

Let A be an abelian category, i : B → A a full abelian subcategory, closed under taking subquotients in A. Suppose that each object M of A admits a finite filtration 0 = M 0 ⊂ M1 ⊂ . . ⊂ M n = M with quotients Mi /Mi−1 all in B. Then BQi : BQ(B) → BQ(A) induces an isomorphism i∗ : Kp (B) → Kp (A) for all p. 8 ([30, Theorem 3, §4]). Let i : E0 → E1 be a full exact subcategory of an exact subcategory E1 , with E0 closed under extensions A SHORT COURSE IN K-THEORY MEXICO CITY MAY, 2002 45 in E1 .

Using this description, Bloch has constructed interesting elements in H 0 (E, K2 ), and Beilinson [4] has constructed analogous elements in H 0 (C, K2 ), where C is a modular curve, and related these elements to values of the Lfunction of C. , Infinite loop spaces. Annals of Mathematics Studies, 90. , 1978. , Solution of the congruence subgroup problem for SLn (n ≥ 3) and Sp2n (n ≥ 2), Inst. Hautes tudes Sci. Publ. Math. 33 1967 59–137. , The Milnor ring of a global field. Algebraic K-theory, II: “Classical” algebraic K-theory and connections with arithmetic (Proc.

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