Remark 63.20.3. Since we have only developed some theory of traces and not of determinants, Theorem 63.20.1 is harder to prove than Theorem 63.20.2. We will only prove the latter, for the former see [SGA4.5]. Observe also that there is no version of this theorem more general for $\mathbf{Z}_\ell $ coefficients since there is no $\ell $-torsion.

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