Motive

Definition

Let $k$ be a field, and let $\mathcal{V}$ be the class of smooth projective varieties over $k$. Let $X, Y \in \mathcal{V}$ with $X$ irreducible of dimension $d$, and let $r \in \Z$. The group of correspondences of degree $r$ is $\operatorname{Corr}^r(X, Y) = CH^{d + r}(X \times Y) \otimes \Q$, where $CH^\bullet$ is the Chow group graded by codimension, that is, algebraic cycles modulo rational equivalence. For general $X$, sum over the irreducible components of $X$. The composition of $f \in \operatorname{Corr}^r(X, Y)$ and $g \in \operatorname{Corr}^s(Y, Z)$ is $$g \circ f = (p_{XZ})_*\big(p_{XY}^* f \cdot p_{YZ}^* g\big) \in \operatorname{Corr}^{r+s}(X, Z),$$ where the $p$ are the projections from $X \times Y \times Z$ and the product is the intersection product.

The category of Chow motives over $k$ with rational coefficients is defined as follows.

1. Its objects, the motives, are triples $(X, p, m)$ with $X \in \mathcal{V}$, $p \in \operatorname{Corr}^0(X, X)$ such that $p \circ p = p$, and $m \in \Z$.
2. Its morphisms are $\Hom\big((X, p, m), (Y, q, n)\big) = q \circ \operatorname{Corr}^{n - m}(X, Y) \circ p$.
3. Composition is the composition of correspondences.

The motive of $X$ is $h(X) = (X, \Delta_X, 0)$, where $\Delta_X$ is the class of the diagonal.