Motive (algebraic geometry)
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High Quality Content by WIKIPEDIA articles! In algebraic geometry, a motive (or sometimes motif, following French usage) denotes refers to 'some essential part of an algebraic variety'. To date, pure motives have been defined, while conjectural mixed motives have not. Pure motives are triples (X, p, m), where X is a smooth projective variety, p : X X is an idempotent correspondence, and m an integer. A morphism from (X, p, m) to (Y, q, n) is given by a correspondence of degree n m. As far as mixed motives, mathematicians are working to find a suitable definition which will then provide a…mehr

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High Quality Content by WIKIPEDIA articles! In algebraic geometry, a motive (or sometimes motif, following French usage) denotes refers to 'some essential part of an algebraic variety'. To date, pure motives have been defined, while conjectural mixed motives have not. Pure motives are triples (X, p, m), where X is a smooth projective variety, p : X X is an idempotent correspondence, and m an integer. A morphism from (X, p, m) to (Y, q, n) is given by a correspondence of degree n m. As far as mixed motives, mathematicians are working to find a suitable definition which will then provide a "universal" cohomology theory. In terms of category theory, it was intended to have a definition via splitting idempotents in a category of algebraic correspondences.