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The thing to keep in mind is that there is 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 framework to underly them.
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2 Remarks |
There are many things one may be interested in for an algebraic variety.
It's interesting to compute its number of points in some finite field.
Its cohomology comes with different structures:
The answer is: people have tried to precisely define this theory for many years.
The current name of this theory is theory of motives.
uf. more explanation here
What is a Motive?
Examples
Each algebraic variety X has a corresponding motive [X], so the simplest examples of motives are:
These 'equations' hold in many situations, namely:
Each motive is graded by degree (for example, the motive [X] is graded from 0 to 2 dim X). Unlikely to usual varieties one can always extract each degree (as it's an image of the whole motive under some of projection), example:
is 1-graded non-trivial motive.The idea
The general idea is that one motive can be realized in different cohomology theories,
One may ask whether there exists some universal theory which embodies all these structures and provides a common ground for equations like
[projective line] = [line]+[point].Definition
A formal definition of motive is: [please clarify]
The result is the category of motives.Remarks
Motives are part of large abstract algebraic geometry program initiated by Alexander Grothendieck. The consistency of theory of motives still reqires some conjectures to be proven and at the present monent there are different definitions of motives. Reading list