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The moving lemma for higher Chow groups. (English) Zbl 0830.14003
For $$V$$ a quasi-projective variety let $$Z(V, \cdot)$$ be the simplicial abelian group defined, in degree $$n$$ by algebraic cycles on $$V \times \Delta^n$$ whose support is meeting all faces properly: the homotopy groups are, by definition, the higher Chow groups of $$V$$. For a given $$X$$ let $$U$$ be a Zariski open subset of $$X$$ and $$Y = X - U$$; we then have a canonical map $$Z(X, \cdot)/Z (Y, \cdot) \to Z (U, \cdot)$$. The “moving lemma” claims that the map above is a homotopy equivalence, yielding the expected long exact sequence of higher Chow groups of $$Y$$, $$X$$ and $$U$$. The basic idea in the proof is to move by blowing up faces but this is achieved only after a combination of extremely delicate simplicial arguments.

##### MSC:
 14C05 Parametrization (Chow and Hilbert schemes) 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 14C99 Cycles and subschemes
##### Keywords:
moving lemma; higher Chow groups