Introduction The theory of motives was created by Grothendieck in the mid-sixties, starting from around 1964. In a letter to Serre dated August, 16 1964 Grothendieck men- tioned for the first time the notion of “motives” see [Col-Se, pages 173 and 275]. At that time this notion was still rather vague for him, but he was already begin- ning to see a precise “yoga” for such a theory (see the letter cited above) and his “motivation”(!) (or at least one of his motivations) must have been the following. In the early sixties Grothendieck, with the help of Artin and Verdier, had devel- oped ´ etale cohomology theory. From that moment on there existed a cohomology theory for every prime number different from the characteristic p of the underly- ing field. Moreover, in characteristic zero there exist also the classical Betti and de Rham theory and for positive characteristic Grothendieck already had the outline for the crystalline cohomology theory. Hence there was an abundance of “good” (so-called Weil) cohomology theories! But all these theories have similar properties, and in characteristic zero there even are comparison theorems between them: the famous de Rham isomorphism theorem between Betti and de Rham theory, and the Artin isomorphism between Betti and ´ etale cohomology. There should be a deeper reason behind this! In order to explain and under- stand this, Grothendieck envisioned a “universal” cohomology theory for algebraic varieties: the theory of motives. Grothendieck expected that there should exist a suitable Q-linear semisimple abelian tensor category with “realization” functors to all Weil cohomology theories.1 The best way to see what Grothendieck had in mind is to quote his own words. In section 16 (les motifs - ou la cœur dans la cœur) of the “En guise d’avant propos” of his “R´ ecoltes et Semailles” [Groth85], Grothendieck writes the following: ... Contrairement ` ce qui se passait en topologie ordinaire, on se trouve donc plac´e l` devant une abondance econcertante de th´ eories cohomologiques diff´ erentes. On avait l’impression tr` es nette qu’en un sens, qui restait d’abord tr` es flou, toutes ces th´ eories devaient ‘revenir au eme’, qu’elles ‘donnaient les emes esultats’. C’est pour parvenir ` exprimer cette intuiton de ‘parent´ e’ entre th´ eories cohomologiques diff´ erentes que j’ai egag´ e la notion de ‘motif’ associ´ e ` une vari´ et´ e alg´ ebrique. Par ce terme j’entends sugg´ erer qu’il s’agit du ‘motif commun’ (ou de la raison com- mune) sous-jacent ` cette multitude d’invariants cohomologiques diff´ erents associ´es ` la vari´ et´ e, ` l’aide de la multitude de toutes les th´ eories cohomologiques possi- bles ` priori. [...] . Ainsi, le motif associ´ e ` une vari´ et´ e alg´ ebrique constituerait 1 (M.) I remember that during a private conversation in October or November 1964 Grothendieck told me that he was now developing a new theory that would finally explain the (similar) behaviour of all the different cohomology theories. v
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