Abstract
‘l’he purpose of this paper is to develop higher algebraic K-theory into a tool for understanding algebraic cycles on a variety. Bloch made the first step: he showed that the group of zero-cycles modulo rational equivalence is Ha(X, L%$) on a nonsingular surface X. Gersten reduced the general statement that H”(X, .%‘J is A”(X), the group of codimension n-cycles modulo rational equivalence, to a conjecture which Quillen proved for nonsingular X. Bloch’s next idea was the relativization of the situation using K-theory. Denote the functor S * H”(X x S, ZJ by &n(X). The question of representability of this functor is interesting. Witness the result of Bloch [4] which says that, for a nonsingular surface X of characteristic zero, the following statements are equivalent:
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