Projections, cycles, and algebraic K-theory

‘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: (I) The formal completion of,cF(X) at 0 is pro-representable. (2) The formal completion of L&“(X) at 0 a g rees with that of the Albanese variety of X, Alb,. (3) The geometric genus, p, = dim H2(X, ox), is zero. The first two statements have the following global analogs: (1’) AZ(X) is finite-dimensional [13]. (2’) P(X) = Z x Alb,(k) (k = base field). These latter are statements about the Chow group of zero-cycles on X, i.e., about the rational points of the functor S@(X). Roitman demonstrated the equivalence of (1’) and (2’), and Mumford showed that they implyp, = 0. Bloch conjectured that AZ(X) is finite-dimensional whenever pII = 0, and proved it for surfaces not of general type [S]; however, his methods yield no grasp of the general situation