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Theorem 64.20.4. Let X/k be as above, let \Lambda be a finite ring with \# \Lambda \in k^* and K\in D_{ctf}(X, \Lambda ). Then R\Gamma _ c(X_{\bar k}, K)\in D_{perf}(\Lambda ) and

\sum _{x\in X(k)}\text{Tr}\left(\pi _ x |_{K_{\bar x}}\right) = \text{Tr}\left(\pi _ X^* |_{R\Gamma _ c(X_{\bar k}, K )}\right).

Proof. Note that we have already proved this (REFERENCE) when \dim X \leq 1. The general case follows easily from that case together with the proper base change theorem. \square


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