Motive (algebraic geometry)

In algebraic geometry the idea of a motive intuitively refers to

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'some essential part of an algebraic variety'. Mathematically, the theory of motives is then the conjectural "universal" cohomology theory for such objects. In terms of category theory, it was intended to have a definition via splitting idempotents in a category of algebraic correspondences. The way ahead for that definition has been blocked for some decades, by the failure to prove the standard conjectures on algebraic cycles. This prevents the category from having 'enough' morphisms, as can currently be shown. While the category of motives was supposed to be the universal Weil cohomology much discussed in the years 1960-1970, that hope for it remains unfulfilled. On the other hand, by a different route, motivic cohomology now has a technically-adequate definition.

Related Topics:
Algebraic variety - Cohomology theory - Category theory - Splitting idempotents - Correspondence - Standard conjectures on algebraic cycles - Universal Weil cohomology - Motivic cohomology

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There is therefore no well-established theory of motives yet. Instead, we know some facts and relationships between them that (as generally accepted among mathematicians) point to the existence of general underlying framework. Some mathematicians prefer the word motif to motive for the singular, following French usage.

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