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Algebraic cycle

In [mathematics] , an ** algebraic cycle ** on an [algebraic variety] * V * is, roughly speaking, a [homology class] on * V * that is represented by a linear combination of [subvarieties] of * V * . Therefore the algebraic cycles on * V * are the part of the [algebraic topology] of * V * that is directly accessible in [algebraic geometry] . With the formulation of some fundamental conjectures in the 1950s and 1960s, the study of algebraic cycles became one of the main objectives of the algebraic geometry of general varieties.

The nature of the difficulties is quite plain: the existence of algebraic cycles is easy to predict, but the methods of construction of them are currently deficient. The major conjectures on algebraic cycles include the [Hodge conjecture] and the [Tate conjecture] . In the search for a proof of the [Weil conjectures] , [Alexander Grothendieck] and [Enrico Bombieri] formulated what are now known as the [standard conjectures of algebraic cycle] theory.

Algebraic cycles have also been shown to be closely connected with [algebraic K-theory] .

For the purposes of a well-working [intersection theory] , one uses various [equivalence relations on algebraic cycles] . Particularly important is the so-called * rational equivalence * . Cycles up to rational equivalence form a graded ring, the [Chow ring] , the multiplication is given by the [intersection product] . Further fundamental relations include * algebraic equivalence * , * numerical equivalence * , and * homological equivalence * . They have (partly conjectural) applications in the theory of [motives] .

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