The Stacks project

Remark 64.16.2. Remarks on Theorem 64.16.1.

1. This formula holds in any dimension. By a dévissage lemma (which uses proper base change etc.) it reduces to the current statement – in that generality.

2. The complex $R\Gamma _ c(X_{\bar k}, K)$ is defined by choosing an open immersion $j : X \hookrightarrow \bar X$ with $\bar X$ projective over $k$ of dimension at most 1 and setting

   $$R\Gamma _ c(X_{\bar k}, K) := R\Gamma (\bar X_{\bar k}, j_!K).$$

   This is independent of the choice of $\bar X$ follows from (insert reference here). We define $H^ i_ c(X_{\bar k}, K)$ to be the $i$th cohomology group of $R\Gamma _ c(X_{\bar k}, K)$.