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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.

14C05 Parametrization (Chow and Hilbert schemes)
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14C99 Cycles and subschemes