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Fix an isomorphism $\tau: \overline{\mathbf{Q}_\ell} \cong \mathbf{C}$ and consider Weil sheaves on a scheme $X$ over $\mathbf{F_q}$.

It is a theorem that a $\tau$-real sheaf $\mathscr{F}$ is $\tau$-mixed, from which it follows that direct summands of $\mathscr{F}$ are also $\tau$-mixed.

Is the converse true? That is, is any $\tau$-mixed sheaf $\mathscr{G}$ always a direct summand of a $\tau$-real sheaf? If so, how is this proven?

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  • $\begingroup$ This is stated to be true on page 37 of these notes. I think it should follow from the result for $\tau$-pure sheaves, which is Lemma 3.14 on page 14. $\endgroup$
    – cnpJj2dwc
    Commented May 4, 2020 at 18:52
  • $\begingroup$ I saw the notes. But I did not see a proof. $\endgroup$
    – Kim
    Commented May 4, 2020 at 22:36

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