Theorem 63.20.4. 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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