##
**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.

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 |