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) is an isomorphism.
\[ Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (L, Rf^!K) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (Rf_!L, K) \]
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 of global derived homs.
\[ R\mathop{\mathrm{Hom}}\nolimits _ X(L, Rf^!K) \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _ Y(Rf_!L, K) \]
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 of quasi-compact and quasi-separated schemes with $f$ separated and of finite type. Then we have $Rf^! \circ Rg_* = Rg'_* \circ R(f')^!$.
\[ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y } \]
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 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 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.
\[ \xymatrix{ D(X_{\acute{e}tale}, \Lambda _2) \ar[r]_{res} & D(X_{\acute{e}tale}, \Lambda _1) \\ D(Y_{\acute{e}tale}, \Lambda _2) \ar[r]^{res} \ar[u]^{Rf^!} & D(Y_{\acute{e}tale}, \Lambda _1) \ar[u]_{Rf^!} } \]
\[ \mathop{\mathrm{Hom}}\nolimits _{\Lambda _2}(K_1 \otimes _{\Lambda _1}^\mathbf {L} \Lambda _2, K_2) = \mathop{\mathrm{Hom}}\nolimits _{\Lambda _1}(K_1, res(K_2)) \]
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