The Stacks project

Theorem 63.14.5. Let $f : X \to Y$ be a separated morphism of finite presentation of quasi-compact and quasi-separated schemes. Let $\Lambda $ be a Noetherian ring. Let $K$ be an object of $D^+_{tors, c}(X_{\acute{e}tale}, \Lambda )$ or of $D_ c(X_{\acute{e}tale}, \Lambda )$ in case $\Lambda $ is torsion. Then $Rf_!K$ has constructible cohomology sheaves, i.e., $Rf_!K$ is in $D^+_{tors, c}(Y_{\acute{e}tale}, \Lambda )$ or in $D_ c(Y_{\acute{e}tale}, \Lambda )$ in case $\Lambda $ is torsion.

Proof. The question is local on $Y$ hence we may and do assume $Y$ is affine. By the induction principle and Lemma 63.10.4 we reduce to the case where $X$ is also affine.

Assume $X$ and $Y$ are affine. Since $X$ is of finite presentation, we can choose a closed immersion $i : X \to \mathbf{A}^ n_ Y$ which is of finite presentation. If $p : \mathbf{A}^ n_ Y \to Y$ denotes the structure morphism, then we see that $Rf_! = Rp_! \circ Ri_!$ by Lemma 63.9.2. By Lemma 63.14.1 we have the result for $Ri_! = i_!$. Hence we may assume $f$ is the projection morphism $\mathbf{A}^ n_ Y \to Y$. Since we can view $f$ as the composition

\[ X = \mathbf{A}^ n_ Y \to \mathbf{A}^{n - 1}_ Y \to \mathbf{A}^{n - 2}_ S \to \ldots \to \mathbf{A}^1_ Y \to Y \]

we may assume $n = 1$.

Assume $Y$ is affine and $X = \mathbf{A}^1_ Y$. Since $Rf_!$ has finite cohomological dimension (Lemma 63.10.2) we may assume $K$ is bounded below. Using the first spectral sequence of Derived Categories, Lemma 13.21.3 (or alternatively using an argument with truncations), we reduce to showing the result of Lemma 63.14.4. $\square$

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