The Stacks project

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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