Definition 45.5.1. Let $k$ be a base field. Let $M = (X, p, m)$ be a Chow motive over $k$. For $i \in \mathbf{Z}$ we define the *$i$th Chow group of $M$* by the formula

## 45.5 Chow groups of motives

We define the Chow groups of a motive as follows.

We have $\mathop{\mathrm{CH}}\nolimits ^ i(h(X)) = \mathop{\mathrm{CH}}\nolimits ^ i(X) \otimes \mathbf{Q}$ if $X$ is a smooth projective scheme over $k$.

Observe that $\mathop{\mathrm{CH}}\nolimits ^ i(-)$ is a functor from $M_ k$ to $\mathbf{Q}$-vector spaces. Indeed, if $c : M \to N$ is a morphism of motives $M = (X, p, m)$ and $N = (Y, q, n)$, then $c$ is a correspondence of degree $n - m$ from $X$ to $Y$ and hence pushforward along $c$ (Section 45.3) is a family of maps

Since $c = q \circ c \circ p$ by definition of morphisms of motives, we see that indeed we obtain

for all $i \in \mathbf{Z}$. This is compatible with compositions of morphisms of motives by Lemma 45.3.1. This functoriality of Chow groups can also be deduced from the following lemma.

Lemma 45.5.2. Let $k$ be a base field. The functor $\mathop{\mathrm{CH}}\nolimits ^ i(-)$ on the category of motives $M_ k$ is representable by $\mathbf{1}(-i)$, i.e., we have

functorially in $M$ in $M_ k$.

**Proof.**
Immediate from the definitions and Lemma 45.3.1.
$\square$

The reader can imagine that we can use Lemma 45.5.2, the Yoneda lemma, and the duality in Lemma 45.4.9 to obtain the following.

Lemma 45.5.3 (Manin). Let $k$ be a base field. Let $c : M \to N$ be a morphism of motives. If for every smooth projective scheme $X$ over $k$ the map $c \otimes 1 : M \otimes h(X) \to N \otimes h(X)$ induces an isomorphism on Chow groups, then $c$ is an isomorphism.

**Proof.**
Any object $L$ of $M_ k$ is a summand of $h(X)(m)$ for some smooth projective scheme $X$ over $k$ and some $m \in \mathbf{Z}$. Observe that the Chow groups of $M \otimes h(X)(m)$ are the same as the Chow groups of of $M \otimes h(X)$ up to a shift in degrees. Hence our assumption implies that $c \otimes 1 : M \otimes L \to N \otimes L$ induces an isomorphism on Chow groups for every object $L$ of $M_ k$. By Lemma 45.5.2 we see that

is an isomorphism for every $L$. Since every object of $M_ k$ has a left dual (Lemma 45.4.10) we conclude that

is an isomorphism for every object $K$ of $M_ k$, see Categories, Lemma 4.43.6. We conclude by the Yoneda lemma (Categories, Lemma 4.3.5). $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)