Motive (algebraic geometry)

EN Texto en inglés Aún no hay artículo en tu idioma: extracto en inglés.

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.