Motive (algebraic Geometry) - The Search For A Universal Cohomology

The Search For A Universal Cohomology

Each algebraic variety X has a corresponding motive , so the simplest examples of motives are:

• = +
• = + +

These 'equations' hold in many situations, namely for de Rham cohomology and Betti cohomology, l-adic cohomology, the number of points over any finite field, and in multiplicative notation for local zeta-functions.

The general idea is that one motive has the same structure in any reasonable cohomology theory with good formal properties; in particular, any Weil cohomology theory will have such properties. There are different Weil cohomology theories, they apply in different situations and have values in different categories, and reflect different structural aspects of the variety in question:

• Betti cohomology is defined for varieties over (subfields of) the complex numbers, it has the advantage of being defined over the integers and is a topological invariant
• de Rham cohomology (for varieties over ℂ) comes with a mixed Hodge structure, it is a differential-geometric invariant
• l-adic cohomology (over any field of characteristic ≠ l) has a canonical Galois group action, i.e. has values in representations of the (absolute) Galois group
• crystalline cohomology

All these cohomology theories share common properties, e.g. existence of Mayer-Vietoris-sequences, homotopy invariance (H*(X)≅H*(X × A1), the product of X with the affine line) and others. Moreover, they are linked by comparison isomorphisms, for example Betti cohomology H*_Betti(X, Z/n) of a smooth variety X over C with finite coefficients is isomorphic to l-adic cohomology with finite coefficients.

The theory of motives is an attempt to find a universal theory which embodies all these particular cohomologies and their structures and provides a framework for "equations" like

= +.

In particular, calculating the motive of any variety X directly gives all the information about the several Weil cohomology theories H*_Betti(X), H*_DR(X) etc.

Beginning with Grothendieck, people have tried to precisely define this theory for many years.