The Stacks project

45.5 Chow groups of motives

We define the Chow groups of a motive as follows.

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

\[ \mathop{\mathrm{CH}}\nolimits ^ i(M) = p\left(\mathop{\mathrm{CH}}\nolimits ^{i + m}(X) \otimes \mathbf{Q}\right) \]

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

\[ c_* : \mathop{\mathrm{CH}}\nolimits ^{i + m}(X) \otimes \mathbf{Q} \longrightarrow \mathop{\mathrm{CH}}\nolimits ^{i + n}(Y) \otimes \mathbf{Q} \]

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

\[ c_* : \mathop{\mathrm{CH}}\nolimits ^ i(M) \to \mathop{\mathrm{CH}}\nolimits ^ i(N) \]

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

\[ \mathop{\mathrm{CH}}\nolimits ^ i(M) = \mathop{\mathrm{Hom}}\nolimits _{M_ k}(\mathbf{1}(-i), M) \]

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

\[ \mathop{\mathrm{Hom}}\nolimits _{M_ k}(\mathbf{1}, M \otimes L) \to \mathop{\mathrm{Hom}}\nolimits _{M_ k}(\mathbf{1}, N \otimes L) \]

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

\[ \mathop{\mathrm{Hom}}\nolimits _{M_ k}(K, M) \to \mathop{\mathrm{Hom}}\nolimits _{M_ k}(K, N) \]

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). $\square$

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