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 ` a ce qui se passait en topologie ordinaire, on se trouve donc plac´e
l` a 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 ` a exprimer cette intuiton de ‘parent´ e’ entre th´ eories cohomologiques
diff´ erentes que j’ai egag´ e la notion de ‘motif’ associ´ e ` a 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 ` a cette multitude d’invariants cohomologiques diff´ erents associ´es
` a la vari´ et´ e, ` a l’aide de la multitude de toutes les th´ eories cohomologiques possi-
bles ` a priori. [...] . Ainsi, le motif associ´ e ` a 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.
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