Lemma 45.4.6. The category $M_ k$ is additive.
Proof. Let $(Y, p, m)$ and $(Z, q, n)$ be motives. If $n = m$, then a direct sum is given by $(Y \amalg Z, p + q, m)$, with obvious notation. Details omitted.
Suppose that $n < m$. Let $X$, $c_2$ be as in Example 45.3.7. Then we consider
\begin{align*} (Z, q, n) & = (Z, q, m) \otimes (\mathop{\mathrm{Spec}}(k), 1, -1) \otimes \ldots \otimes (\mathop{\mathrm{Spec}}(k), 1, -1) \\ & \cong (Z, q, m) \otimes (X, c_2, 0) \otimes \ldots \otimes (X, c_2, 0) \\ & \cong (Z \times X^{m - n}, q \otimes c_2 \otimes \ldots \otimes c_2, m) \end{align*}
where we have used Lemma 45.4.4. This reduces us to the case discussed in the first paragraph. $\square$
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