In algebraic geometry, a motive (or sometimes motif, following French usage) is an abstract object introduced by Alexander Grothendieck in the 1960s as part of a proposed universal cohomology theory for algebraic varieties. The idea is that theories such as Betti cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology should arise as different realizations of the same underlying object. Thus, a motive of a variety is meant to capture the part of its geometry common to these cohomology theories. For smooth varieties, pure motives can be constructed from algebraic correspondences and idempotent projections. The Chow motives give one example. For more general varieties, the (partially conjectural) theory of mixed motives extends this picture to relate motives to motivic cohomology and algebraic K-theory. Vladimir Voevodsky constructed triangulated categories of motives that provide much of the formalism expected of mixed motives. The theory of motives is connected to algebraic cycles, Weil cohomology, and the study of motivic Galois groups. It also provides a unifying framework for open problems such as the Hodge conjecture and Tate conjecture.
Introduction 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. From another viewpoint, motives continue the sequence of generalizations from rational functions on varieties to divisors on varieties to Chow groups of varieties. The generalization happens in more than one direction, since motives can be considered with respect to more types of equivalence than rational equivalence. The admissible equivalences are given by the definition of an adequate equivalence relation.
Grothendieck and Deligne formulations In the formulation of Grothendieck for smooth projective varieties, a motive is a triple ( X , p , m ) {\displaystyle (X,p,m)} , where X {\displaystyle X} is a smooth projective variety, p : X ⊢ X {\displaystyle p:X\vdash X} is an idempotent correspondence, and m an integer; however, such a triple contains almost no information outside the context of Grothendieck's category of pure motives, where a morphism from ( X , p , m ) {\displaystyle (X,p,m)} to ( Y , q , n ) {\displaystyle (Y,q,n)} is given by a correspondence of degree n − m {\displaystyle n-m} . 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
( M B , M D R , M A f , M cris , p , comp D R , B , comp A f , B , comp cris p , D R , W , F ∞ , F , ϕ , ϕ p ) {\displaystyle \left(M_{B},M_{\mathrm {DR} },M_{\mathbb {A} ^{f}},M_{\operatorname {cris} ,p},\operatorname {comp} _{\mathrm {DR} ,B},\operatorname {comp} _{\mathbb {A} ^{f},B},\operatorname {comp} _{\operatorname {cris} p,\mathrm {DR} },W,F_{\infty },F,\phi ,\phi _{p}\right)}
consisting of modules
M B , M D R , M A f , M cris , p {\displaystyle M_{B},M_{\mathrm {DR} },M_{\mathbb {A} ^{f}},M_{\operatorname {cris} ,p}}
over the rings
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