## Zero-cycles and K-theory on singular varieties.(English)Zbl 0635.14007

Algebraic geometry, Proc. Summer. Res. Inst., Brunswick/Maine 1985, part. 2, Proc. Symp. Pure Math. 46, 451-462 (1987).
[For the entire collection see Zbl 0626.00011.]
A theorem of A. Rojtman [Ann. Math., II. Ser. 111, 553-569 (1980; Zbl 0504.14006)] says that for a smooth projective variety X over an algebraically closed field, the torsion subgroup of the group of zero- cycles modulo rational equivalence, $$CH_ 0(X)_{tors}$$, is naturally isomorphic to the torsion subgroup of the Albanese variety of X, $$Alb(X)_{tors}$$. This result was generalized by the author [Am. J. Math. 107, 737-757 (1985; Zbl 0579.14007)] for the case of a projective variety X, smooth in codimension one (modulo p-torsion in characteristic $$p,$$ $$p>0)$$, where $$CH_ 0(X)_{tors}$$ had to be replaced by the torsion part of $$CH_ 0(X,X_{\sin g})$$, the quotient of the free abelian group on the smooth points of X and the subgroup generated by cycles of the form $$i_{C\quad *}((f))$$ with C a closed reduced, irreducible curve on X, not meeting $$X_{\sin g}$$, and f a rational function on C.
In the general case one is led to define a relative Chow group $$CH_ 0(X,Y)$$, where Y is closed in X and contains $$X_{\sin g}$$. The definition was given in another article and is not repeated here. Now the main result of the paper under review is that for an affine variety X over an algebraically closed field $$CH_ 0(X,Y)$$ is torsion free (modulo p-torsion in characteristic $$p>0).$$
As a corollary one obtains an injective map $$CH_ 0(X,Y)\to K_ 0(X)$$, whose image is isomorphic to the subgroup of $$K_ 0$$(X) generated by the residue fields of the smooth points of X.
Reviewer: W.W.J.Hulsbergen

### MSC:

 14C15 (Equivariant) Chow groups and rings; motives 14B05 Singularities in algebraic geometry 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 14C25 Algebraic cycles

### Citations:

Zbl 0626.00011; Zbl 0504.14006; Zbl 0579.14007