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

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

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