Definition 63.12.1. Let $X$ be a separated scheme of finite type over a field $k$. Let $\Lambda $ be a ring. Let $K$ be an object of $D^+_{tors}(X_{\acute{e}tale}, \Lambda )$ or of $D(X_{\acute{e}tale}, \Lambda )$ in case $\Lambda $ is torsion. The cohomology of $K$ with compact support or the compactly supported cohomology of $K$ is
\[ R\Gamma _ c(X, K) = R\Gamma (\mathop{\mathrm{Spec}}(k), Rf_!K) \]
where $f : X \to \mathop{\mathrm{Spec}}(k)$ is the structure morphism. We will write $H^ i_ c(X, K) = H^ i(R\Gamma _ c(X, K))$.
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