Motive (algebraic geometry)

In algebraic geometry, motives (or sometimes motifs, following French usage) is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology. Philosophically, a "motif" is the "cohomology essence" of a variety. In the formulation of Grothendieck for smooth projective varieties, a motive is a triple , where X is a smooth projective variety, is an idempotent correspondence, and m an integer, however, such a triple contains almost no information outside the context of Grothendieck's of pure motives, where a morphism from to is given by a correspondence of degree . A more object-focused approach is taken by Pierre Deligne in Le Groupe Fondamental de la Droite Projective Moins Trois Points. In that article, a motive is a "system of realisations" – that is, a tuple consisting of modules over the rings respectively, various comparison isomorphisms between the obvious base changes of these modules, filtrations , a -action on and a "Frobenius" automorphism of . This data is modeled on the cohomologies of a smooth projective -variety and the structures and compatibilities they admit, and gives an idea about what kind of information is contained in a motive. The theory of motives was originally conjectured as an attempt to unify a rapidly multiplying array of cohomology theories, including Betti cohomology, de Rham cohomology, l-adic cohomology, and crystalline cohomology. The general hope is that equations like [projective line] = [line] + [point] [projective plane] = [plane] + [line] + [point] can be put on increasingly solid mathematical footing with a deep meaning. Of course, the above equations are already known to be true in many senses, such as in the sense of CW-complex where "+" corresponds to attaching cells, and in the sense of various cohomology theories, where "+" corresponds to the direct sum.

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