Main Page | See live article | Alphabetical index In algebraic geometry the idea of motive intuitively refers to 'some essential part of variety'. Mathematically, the theory of motives is the universal cohomology theory. 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. Table of contents 1 What is Motive? 1.1 Examples 1.2 The idea 1.3 Definition 2 Remarks 2.4 Reading list What is Motive? Examples Each algebraic variety X has corresponding motive [X], so the simplest examples of motives are: [point] [projective line] = [point] + [line] [projective plane] = [plane] + [line] + [point] These 'equations' hold in many situations, namely: for de Rham cohomology and Betti cohomology over complex numbers for étale cohomology over other fields for number of points over any finite field for conducttor (which is of course trivial for above) Each motive is graded by degree (for example 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: h = [elliptic curve] - [line] - [point] is 1-graded non-trivial motive. The idea The general idea is that one motive can be realized in different cohomology theories, 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 come with different structures: Betti cohomology have the advantage of being defined over integer numbers de Rham cohomology come with mixed Hodge structure étale cohomology have canonical Galois group action add here conductor and other stuff... One may ask whether there exists s universal theory which embodies all these structures and provides common ground for equations like [projective line] = [line]+[point]. The answer is: people try to precisely define this theory for many years. The current name of this theory is theory of motives. uf. more explanation here Definition A formal definition of motive is: [please clarify] consider a category of algebraic varieties over some field k with correspondencess as morphisms, make all exact triples into sums and add all images of projectors to make it abelian. 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 math.QA/9904055: Maxim Kontsevich Operads and motives in deformation quantization math.AG/9911179: A.Craw An introduction to motivic integration (preprint, 1992): Alexander Beilinson, Pierre Deligne Motivic polylogarithm and Zagier conjecture This article is from Wikipedia. All text is available under the terms of the GNU Free Documentation License.