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
- = +
- = + +
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 admissiable equivalences are given by the definition of an adequate equivalence relation.
Read more about this topic: Motive (algebraic Geometry)
Famous quotes containing the word introduction:
“For the introduction of a new kind of music must be shunned as imperiling the whole state; since styles of music are never disturbed without affecting the most important political institutions.”
—Plato (c. 427–347 B.C.)
“We used chamber-pots a good deal.... My mother ... loved to repeat: “When did the queen reign over China?” This whimsical and harmless scatological pun was my first introduction to the wonderful world of verbal transformations, and also a first perception that a joke need not be funny to give pleasure.”
—Angela Carter (1940–1992)
“Do you suppose I could buy back my introduction to you?”
—S.J. Perelman, U.S. screenwriter, Arthur Sheekman, Will Johnstone, and Norman Z. McLeod. Groucho Marx, Monkey Business, a wisecrack made to his fellow stowaway Chico Marx (1931)