I'm now working on Kiehl and Weissauer's book Weil Conjectures, Perverse Sheaves and l’adic Fourier Transform. All sheaves mensioned are on the small étale site.
Consider an algebraic scheme $X_0 $ over some finite field $k$ of char=$p$, with algebraic closure $\overline{k}$. In I.2, we fix a prime $\ell\neq p$ and an isomorphism $$\tau:\overline{\mathbb{Q}_{\ell}}\cong \mathbb{C} $$ (that's from the fact that all algebraic closed extensions with a transcendental bases of same order are isom), and then define a sheaf $G_0$ over $X_0$ to be $\tau$-pure of weight $w$ if for all geometric point $\bar{x}$, the arithmetic frobenius $F_x$ on ${G_{0}}_{\overline{x}}$ all eigenvalues $\alpha$ s.t. $$|\tau(\alpha)|=(\#k)^{w/2}.$$
Now remark 2.2 states a conclusion that left me confused:
Question: Let $G_0$ be pure for any $\tau$, then why we can find a $b\in \overline{\mathbb{Q}_{\ell}}$ that
$$G_0=F_0\otimes L_b,$$
where $F_0$ is pure for any $\tau$ and of weight $w$ independent of $\tau$, $L_b$ is the 'rank-1' sheaf defined by characteristic (or Galois representation) $$\phi_b:Gal(k)\rightarrow \overline{\mathbb{Q}_{\ell}}^*$$
$$1\mapsto b.$$