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
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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