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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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