Remark 64.15.10. Let us try to illustrate the content of the formula of Lemma 64.15.8. Suppose that $\Lambda $, viewed as a trivial $\Gamma $-module, admits a finite resolution $ 0\to P_ r\to \ldots \to P_1 \to P_0\to \Lambda \to 0 $ by some $\Lambda [\Gamma ]$-modules $P_ i$ which are finite and projective as $\Lambda [G]$-modules. In that case

and

Therefore, Lemma 64.15.8 says

This can be interpreted as a point count on the stack $BG$. If $\Lambda = \mathbf{F}_\ell $ with $\ell $ prime to $\# G$, then $H_*(G, \Lambda )$ is $\mathbf{F}_\ell $ in degree 0 (and $0$ in other degrees) and the formula reads

This is in some sense a “trivial” trace formula for $G$. Later we will see that (64.14.3.1) can in some cases be viewed as a highly nontrivial trace formula for a certain type of group, see Section 64.30.

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