The Stacks project

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.

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.

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 √Čtale 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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