Lemma 63.7.1. Let f : X \to Y be a locally quasi-finite morphism of schemes. Let \Lambda be a ring. The functors f_! and f^! of Definition 63.4.4 and Lemma 63.6.1 induce adjoint functors f_! : D(X_{\acute{e}tale}, \Lambda ) \to D(Y_{\acute{e}tale}, \Lambda ) and Rf^! : D(Y_{\acute{e}tale}, \Lambda ) \to D(X_{\acute{e}tale}, \Lambda ) on derived categories.
63.7 Derived upper shriek for locally quasi-finite morphisms
We can take the derived versions of the functors in Section 63.6 and obtain the following.
In the separated case the functor f_! is defined in Section 63.3.
Proof. This follows immediately from Derived Categories, Lemma 13.30.3, the fact that f_! is exact (Lemma 63.4.5) and hence Lf_! = f_! and the fact that we have enough K-injective complexes of \Lambda -modules on Y_{\acute{e}tale} so that Rf^! is defined. \square
Remark 63.7.2. Let f : X \to Y be a locally quasi-finite morphism of schemes. Let \Lambda be a ring. The functor f_! : D(X_{\acute{e}tale}, \Lambda ) \to D(Y_{\acute{e}tale}, \Lambda ) of Lemma 63.7.1 sends complexes with torsion cohomology sheaves to complexes with torsion cohomology sheaves. This is immediate from the description of the stalks of f_!, see Lemma 63.4.5.
Lemma 63.7.3. Let X be a scheme. Let X = U \cup V with U and V open. Let \Lambda be a ring. Let K \in D(X_{\acute{e}tale}, \Lambda ). There is a distinguished triangle
j_{U \cap V!}K|_{U \cap V} \to j_{U!}K|_ U \oplus j_{V!}K|_ V \to K \to j_{U \cap V!}K|_{U \cap V}[1]
in D(X_{\acute{e}tale}, \Lambda ) with obvious notation.
Proof. Since the restriction functors and the lower shriek functors we use are exact, it suffices to show for any abelian sheaf \mathcal{F} on X_{\acute{e}tale} the sequence
0 \to j_{U \cap V!}\mathcal{F}|_{U \cap V} \to j_{U!}\mathcal{F}|_ U \oplus j_{V!}\mathcal{F}|_ V \to \mathcal{F} \to 0
is exact. This can be seen by looking at stalks. \square
Lemma 63.7.4. Let X be a scheme. Let Z \subset X be a closed subscheme and let U \subset X be the complement. Denote i : Z \to X and j : U \to X the inclusion morphisms. Let \Lambda be a ring. Let K \in D(X_{\acute{e}tale}, \Lambda ). There is a distinguished triangle
j_!j^{-1}K \to K \to i_*i^{-1}K \to j_!j^{-1}K[1]
in D(X_{\acute{e}tale}, \Lambda ).
Proof. Immediate consequence of Étale Cohomology, Lemma 59.70.8 and the fact that the functors j_!, j^{-1}, i_*, i^{-1} are exact and hence their derived versions are computed by applying these functors to any complex of sheaves representing K. \square
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