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

Levine, Marc; Weibel, Chuck

J. Reine Angew. Math. 359, 106-120 (1985).

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

Keywords:

\(K_ 0\); \(G_ 0\); relative Chow group; singular variety; rational equivalence; complete intersections