## Zero cycles and complete intersections on singular varieties.(English)Zbl 0555.14004

We construct a relative Chow group $$CH_ 0(X,Y)$$ associated to a singular variety X with a closed subset Y containing the singular locus of X. This comes equipped with a cycle map $$\gamma: X-Y\to CH_ 0(X,Y),$$ and our main geometric result is that the relation of rational equivalence is $$\sigma$$-closed. This is, $$\gamma^{-1}(0)$$ is a countable union of closed subsets of X-Y. We apply this to the case in which $$X=Spec(A)$$ is a singular affine surface and obtain our main algebraic result: the set of maximal ideals of A which are complete intersections is $$\sigma$$-closed in the set of all regular maximal ideals of A.

### MSC:

 14C25 Algebraic cycles 14M10 Complete intersections 14C05 Parametrization (Chow and Hilbert schemes) 14C15 (Equivariant) Chow groups and rings; motives 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 14B05 Singularities in algebraic geometry
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