Zero Cycles and Projective Modules

Abstract

Let A be a reduced affine k-algebra of dimension n over an algebraically closed field k. Let FnKo(A) denote the sub-group of Ko(A) generated by the images of all the residue fields of all smooth maximal ideals of height n. For a module of finite projective dimension M, let (M) denote the image of M in Ko(A). For a projective A-module P of rank n, we define the nth Chern class of P to be: Cn(P) = E(-l)i(AiP*), where P* is the dual of P. Suppose FnKo(A) has no (n 1)! torsion; then our main result (Th. 3.8) is that P has a free direct summand of rank one if and only if Cn(P) = 0. When the characteristic of k is zero or A is normal and n > 3, it is known that FnKo(A) is torsion-free ([Le], [Sr]). Hence our theorem is applicable in these cases. Also when A is regular, FnKo(A) coincides with CHn(X) the Chow group of zero cycles of Spec A and Cn(P) coincides with the usual nth Chern class as defined by Grothendieck (see [Fu]). When n < 3 and A is regular, this result was proved in [MKM]. When n = 3 and the characteristic of k is not equal to two, this result in [MKM] was extended to the singular case by M. Levine. In this paper, we extend the results of [MKM] to all dimensions. With the assumption stated above on torsion in FnKo(A), we first show that (Cor. 3.4) if I C A is a local complete intersection of height n, then (A/I) is zero in Ko(A) if and only if I is a complete intersection. Corollary 3.4 together with a result of Mohan Kumar ([MK], Cor. 1.9 here) at once gives Theorem 3.8. In Section 1, we give some preliminaries and generalize results in [MK2]. These results are crucially used in the rest of the paper. The basic ideas in Section 1 are all taken from [MK2]. In Section 2, we prove Theorem 2.2, which strengthens a result of Boratynski [Bo] for local complete intersections with trivial co-normal bundle. Theorem 2.2 is one of the crucial ingredients in the proofs of the results in Section 3. As an amusing application of Theorem 2.2, we give a new proof of a theorem of Srinivas ([Sr]) about torsion in zero cycles