In this chapter we discuss Weil cohomology theories for smooth projective schemes over a base field. Briefly, for us such a cohomology theory $H^*$ is one which has Künneth, Poincaré duality, and cycle classes (with suitable compatibilities). We warn the reader that there is no universal agreement in the literature as to what constitutes a “Weil cohomology theory”.
Before reading this chapter the reader should take a look at Categories, Section 4.42 and Homology, Section 12.17 where we define (symmetric) monoidal categories and we develop just enough basic language concerning these categories for the needs of this chapter. Equipped with this language we construct in Section 45.3 the symmetric monoidal graded category whose objects are smooth projective schemes and whose morphisms are correspondences. In Section 45.4 we add images of projectors and invert the Lefschetz motive in order to obtain the symmetric monoidal Karoubian category $M_ k$ of Chow motives. This category comes equipped with a contravariant functor
As we will see below, a key property of a Weil cohomology theory is that it factors over $h$.
First, in the case of an algebraically closed base field, we define what we call a “classical Weil cohomology theory”, see Section 45.7. This notion is the same as the notion introduced in [Section 1.2, Kleiman-cycles] and agrees with the notion introduced in [page 65, Kleiman-motives]. However, our notion does not a priori agree with the notion introduced in [page 10, Kleiman-standard] because there the author adds two Lefschetz type axioms and it isn't known whether any classical Weil cohomology theory as defined in this chapter satisfies those axioms. At the end of Section 45.7 we show that a classical Weil cohomology theory is of the form $H^* = G \circ h$ where $G$ is a symmetric monoidal functor from $M_ k$ to the category of graded vector spaces over the coefficient field of $H^*$.
In Section 45.8 we prove a couple of lemmas on cycle groups over non-closed fields which will be used in discussing Weil cohomology theories on smooth projective schemes over arbitrary fields.
Our motivation for our axioms of a Weil cohomology theory $H^*$ over a general base field $k$ are the following
$H^* = G \circ h$ for a symmetric monoidal functor $G$ from $M_ k$ to the category of graded vector spaces over the coefficient field $F$ of $H^*$,
$G$ should send the Tate motive (inverse of the Lefschetz motive) to a $1$-dimensional vector space $F(1)$ sitting in degree $-2$,
when $k$ is algebraically closed we should recover the notion discussion in Section 45.7 up to choosing a basis element of $F(1)$.
In the final Section 45.14 we detail an alternative approach to Weil cohomology theories, using a first Chern class map instead of cycle classes. It is this approach that will be most suited for proving that certain cohomology theories are Weil cohomology theories in later chapters, see de Rham Cohomology, Section 50.22.
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