Lemma 63.11.1. Let f : X \to Y be a finite type separated morphism of quasi-compact and quasi-separated schemes. Let \Lambda be a torsion coefficient ring. The functor Rf_! : D(X_{\acute{e}tale}, \Lambda ) \to D(Y_{\acute{e}tale}, \Lambda ) has a right adjoint Rf^! : D(Y_{\acute{e}tale}, \Lambda ) \to D(X_{\acute{e}tale}, \Lambda ).
63.11 Derived upper shriek
We obtain Rf^! by a Brown representability theorem.
Proof. This follows from Injectives, Proposition 19.15.2 and Lemma 63.10.1 above. \square
Lemma 63.11.2. Let f : X \to Y be a separated quasi-finite morphism of quasi-compact and quasi-separated schemes. Let \Lambda be a torsion coefficient ring. The functor Rf^! : D(Y_{\acute{e}tale}, \Lambda ) \to D(X_{\acute{e}tale}, \Lambda ) of Lemma 63.11.1 is the same as the functor Rf^! of Lemma 63.7.1.
Proof. Follows from uniqueness of adjoints as Rf_! = f_! by Lemma 63.10.3. \square
Lemma 63.11.3. Let j : U \to X be a separated étale morphism of quasi-compact and quasi-separated schemes. Let \Lambda be a torsion coefficient ring. The functor Rj^! : D(X_{\acute{e}tale}, \Lambda ) \to D(U_{\acute{e}tale}, \Lambda ) is equal to j^{-1}.
Proof. This is true because both Rj^! and j^{-1} are right adjoints to Rj_! = j_!. See for example Lemmas 63.11.2 and 63.6.2. \square
Lemma 63.11.4. Let f : X \to Y be a finite type separated morphism of quasi-compact and quasi-separated schemes. Let \Lambda be a torsion ring. The functor Rf^! sends D^+(Y_{\acute{e}tale}, \Lambda ) into D^+(X_{\acute{e}tale}, \Lambda ). More precisely, there exists an integer N \geq 0 such that if K \in D(Y_{\acute{e}tale}, \Lambda ) has H^ i(K) = 0 for i < a then H^ i(Rf^!K) = 0 for i < a - N.
Proof. Let N be the integer found in Lemma 63.10.2. By construction, for K \in D(Y_{\acute{e}tale}, \Lambda ) and L \in \in D(X_{\acute{e}tale}, \Lambda ) we have \mathop{\mathrm{Hom}}\nolimits _ X(L, Rf^!K) = \mathop{\mathrm{Hom}}\nolimits _ Y(Rf_!L, K). Suppose H^ i(K) = 0 for i < a. Then we take L = \tau _{\leq a - N - 1}Rf^!K. By Lemma 63.10.2 the complex Rf_!L has vanishing cohomology sheaves in degrees \leq a - 1. Hence \mathop{\mathrm{Hom}}\nolimits _ Y(Rf_!L, K) = 0 by Derived Categories, Lemma 13.27.3. Hence the canonical map \tau _{\leq a - N - 1}Rf^!K \to Rf^!K is zero which implies H^ i(Rf^!K) = 0 for i \leq a - N - 1. \square
Let f : X \to Y be a separated finite type morphism of quasi-separated and quasi-compact schemes. Let \Lambda be a torsion coefficient ring. For every K \in D(Y_{\acute{e}tale}, \Lambda ) and L \in D(X_{\acute{e}tale}, \Lambda ) we obtain a canonical map
63.11.4.1
\begin{equation} \label{more-etale-equation-sheafy-trace} Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (L, Rf^!K) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (Rf_!L, K) \end{equation}
Namely, this map is constructed as the composition
Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (L, Rf^!K) \to R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (Rf_!L, Rf_!Rf^!K) \to R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (Rf_!L, K)
where the first arrow is Remark 63.10.9 and the second arrow is the counit Rf_!Rf^!K \to K of the adjunction.
Lemma 63.11.5. Let f : X \to Y be a separated finite type morphism of quasi-compact and quasi-separated schemes. Let \Lambda be a torsion ring. For every K \in D(Y_{\acute{e}tale}, \Lambda ) and L \in D(X_{\acute{e}tale}, \Lambda ) the map (63.11.4.1)
Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (L, Rf^!K) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (Rf_!L, K)
is an isomorphism.
Proof. To prove the lemma we have to show that for any M \in D(Y_{\acute{e}tale}, \Lambda ) the map (63.11.4.1) induces an bijection
\mathop{\mathrm{Hom}}\nolimits _ Y(M, Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (L, Rf^!K)) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _ Y(M, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (Rf_!L, K))
To see this we use the following string of equalities
\begin{align*} \mathop{\mathrm{Hom}}\nolimits _ Y(M, Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (L, Rf^!K)) & = \mathop{\mathrm{Hom}}\nolimits _ X(f^{-1}M, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (L, Rf^!K)) \\ & = \mathop{\mathrm{Hom}}\nolimits _ X(f^{-1}M \otimes _\Lambda ^\mathbf {L} L, Rf^!K) \\ & = \mathop{\mathrm{Hom}}\nolimits _ Y(Rf_!(f^{-1}M \otimes _\Lambda ^\mathbf {L} L), K) \\ & = \mathop{\mathrm{Hom}}\nolimits _ Y(M \otimes _\Lambda ^\mathbf {L} Rf_!L, K) \\ & = \mathop{\mathrm{Hom}}\nolimits _ Y(M, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (Rf_!L, K)) \end{align*}
The first equality holds by Cohomology on Sites, Lemma 21.19.1. The second equality by Cohomology on Sites, Lemma 21.35.2. The third equality by construction of Rf^!. The fourth equality by Lemma 63.10.7 (this is the important step). The fifth by Cohomology on Sites, Lemma 21.35.2. \square
Lemma 63.11.6. Let f : X \to Y be a separated finite type morphism of quasi-separated and quasi-compact schemes. Let \Lambda be a torsion ring. For every K \in D(Y_{\acute{e}tale}, \Lambda ) and L \in D(X_{\acute{e}tale}, \Lambda ) the map (63.11.4.1) induces an isomorphism
R\mathop{\mathrm{Hom}}\nolimits _ X(L, Rf^!K) \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _ Y(Rf_!L, K)
of global derived homs.
Proof. By the construction in Cohomology on Sites, Section 21.36 we have
R\mathop{\mathrm{Hom}}\nolimits _ X(L, Rf^!K) = R\Gamma (X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (L, Rf^!K)) = R\Gamma (Y, Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (L, Rf^!K))
(the second equality by Leray) and
R\mathop{\mathrm{Hom}}\nolimits _ Y(Rf_!L, K) = R\Gamma (Y, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (Rf_!L, K))
Thus the lemma is a consequence of Lemma 63.11.5. \square
Lemma 63.11.7. Consider a cartesian square
\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y }
of quasi-compact and quasi-separated schemes with f separated and of finite type. Then we have Rf^! \circ Rg_* = Rg'_* \circ R(f')^!.
Proof. By uniqueness of adjoint functors this follows from base change for derived lower shriek: we have g^{-1} \circ Rf_! = Rf'_! \circ (g')^{-1} by Lemma 63.9.4. \square
Remark 63.11.8. Let \Lambda _1 \to \Lambda _2 be a homomorphism of torsion rings. Let f : X \to Y be a separated finite type morphism of quasi-compact and quasi-separated schemes. The diagram
\xymatrix{ D(X_{\acute{e}tale}, \Lambda _2) \ar[r]_{res} & D(X_{\acute{e}tale}, \Lambda _1) \\ D(Y_{\acute{e}tale}, \Lambda _2) \ar[r]^{res} \ar[u]^{Rf^!} & D(Y_{\acute{e}tale}, \Lambda _1) \ar[u]_{Rf^!} }
commutes where res is the “restriction” functor which turns a \Lambda _2-module into a \Lambda _1-module using the given ring map. This holds by uniquenss of adjoints, the second commutative diagram of Remark 63.10.8 and because we have
\mathop{\mathrm{Hom}}\nolimits _{\Lambda _2}(K_1 \otimes _{\Lambda _1}^\mathbf {L} \Lambda _2, K_2) = \mathop{\mathrm{Hom}}\nolimits _{\Lambda _1}(K_1, res(K_2))
This equality either for objects living over X_{\acute{e}tale} or on Y_{\acute{e}tale} is a very special case of Cohomology on Sites, Lemma 21.19.1.
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