The Stacks project

Lemma 48.32.1. The functor $Rf_!$ is, up to isomorphism, independent of the choice of the compactification.

Proof. Consider the category of compactifications of $X$ over $Y$, which is cofiltered according to More on Flatness, Theorem 38.33.8 and Lemmas 38.32.1 and 38.32.2. To every choice of a compactification

\[ j : X \to \overline{X},\quad \overline{f} : \overline{X} \to Y \]

the construction above associates the functor $R\overline{f}_* \circ Rj_!$. Suppose given a morphism $g : \overline{X}_1 \to \overline{X}_2$ between compactifications $j_ i : X \to \overline{X}_ i$ over $Y$. Then we get an isomorphism

\[ R\overline{f}_{2, *} \circ Rj_{2, !} = R\overline{f}_{2, *} \circ Rg_* \circ j_{1, !} = R\overline{f}_{1, *} \circ Rj_{1, !} \]

using Lemma 48.31.2 in the first equality. In this way we see our functor is independent of the choice of compactification up to isomorphism. $\square$

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