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A cycle on an arbitrary algebraic variety (or scheme) X is a finite formal sum Σ n V [V] of (irreducible) subvarieties of X, with integer coefficients. A rational function r on any subvariety of X determines a cycle [div(r)]. Cycles differing by a sum of such cycles are defined to be rationally equivalent. Alternatively, rational equivalence is generated by cycles of the form [V(0)] − [V(∞)] for subvarieties V of X × ℙ1 which project dominantly to ℙ1. The group of rational equivalence classes on X is denoted A * X.
KeywordsExact Sequence Irreducible Component Local Ring Rational Equivalence Abelian Variety
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