Lemma 86.7.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated algebraic spaces over $S$. The map $Lf^*K \otimes ^\mathbf {L}_{\mathcal{O}_ X} a(L) \to a(K \otimes _{\mathcal{O}_ Y}^\mathbf {L} L)$ defined above for $K, L \in D_\mathit{QCoh}(\mathcal{O}_ Y)$ is an isomorphism if $K$ is perfect. In particular, (86.7.0.1) is an isomorphism if $K$ is perfect.
86.7 Right adjoint of pushforward and pullback
Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated algebraic spaces over $S$. Let $a$ be the right adjoint of pushforward as in Lemma 86.3.1. For $K, L \in D_\mathit{QCoh}(\mathcal{O}_ Y)$ there is a canonical map
\[ Lf^*K \otimes ^\mathbf {L}_{\mathcal{O}_ X} a(L) \longrightarrow a(K \otimes _{\mathcal{O}_ Y}^\mathbf {L} L) \]
Namely, this map is adjoint to a map
\[ Rf_*(Lf^*K \otimes ^\mathbf {L}_{\mathcal{O}_ X} a(L)) = K \otimes ^\mathbf {L}_{\mathcal{O}_ Y} Rf_*(a(L)) \longrightarrow K \otimes ^\mathbf {L}_{\mathcal{O}_ Y} L \]
(equality by Derived Categories of Spaces, Lemma 75.20.1) for which we use the trace map $Rf_*a(L) \to L$. When $L = \mathcal{O}_ Y$ we obtain a map
86.7.0.1
\begin{equation} \label{spaces-duality-equation-compare-with-pullback} Lf^*K \otimes ^\mathbf {L}_{\mathcal{O}_ X} a(\mathcal{O}_ Y) \longrightarrow a(K) \end{equation}
functorial in $K$ and compatible with distinguished triangles.
Proof. Let $K^\vee $ be the “dual” to $K$, see Cohomology on Sites, Lemma 21.48.4. For $M \in D_\mathit{QCoh}(\mathcal{O}_ X)$ we have
\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ Y)}(Rf_*M, K \otimes ^\mathbf {L}_{\mathcal{O}_ Y} L) & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ Y)}( Rf_*M \otimes ^\mathbf {L}_{\mathcal{O}_ Y} K^\vee , L) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}( M \otimes ^\mathbf {L}_{\mathcal{O}_ X} Lf^*K^\vee , a(L)) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(M, Lf^*K \otimes ^\mathbf {L}_{\mathcal{O}_ X} a(L)) \end{align*}
Second equality by the definition of $a$ and the projection formula (Cohomology on Sites, Lemma 21.50.1) or the more general Derived Categories of Spaces, Lemma 75.20.1. Hence the result by the Yoneda lemma. $\square$
Lemma 86.7.2. Suppose we have a diagram (86.4.0.1). Let $K \in D_\mathit{QCoh}(\mathcal{O}_ Y)$. The diagram commutes where the horizontal arrows are the maps (86.7.0.1) for $K$ and $Lg^*K$ and the vertical maps are constructed using Cohomology on Sites, Remark 21.19.3 and (86.4.1.1).
\[ \xymatrix{ L(g')^*(Lf^*K \otimes ^\mathbf {L}_{\mathcal{O}_ X} a(\mathcal{O}_ Y)) \ar[r] \ar[d] & L(g')^*a(K) \ar[d] \\ L(f')^*Lg^*K \otimes _{\mathcal{O}_{X'}}^\mathbf {L} a'(\mathcal{O}_{Y'}) \ar[r] & a'(Lg^*K) } \]
Proof. In this proof we will write $f_*$ for $Rf_*$ and $f^*$ for $Lf^*$, etc, and we will write $\otimes $ for $\otimes ^\mathbf {L}_{\mathcal{O}_ X}$, etc. Let us write (86.7.0.1) as the composition
\begin{align*} f^*K \otimes a(\mathcal{O}_ Y) & \to a(f_*(f^*K \otimes a(\mathcal{O}_ Y))) \\ & \leftarrow a(K \otimes f_*a(\mathcal{O}_ K)) \\ & \to a(K \otimes \mathcal{O}_ Y) \\ & \to a(K) \end{align*}
Here the first arrow is the unit $\eta _ f$, the second arrow is $a$ applied to Cohomology on Sites, Equation (21.50.0.1) which is an isomorphism by Derived Categories of Spaces, Lemma 75.20.1, the third arrow is $a$ applied to $\text{id}_ K \otimes \text{Tr}_ f$, and the fourth arrow is $a$ applied to the isomorphism $K \otimes \mathcal{O}_ Y = K$. The proof of the lemma consists in showing that each of these maps gives rise to a commutative square as in the statement of the lemma. For $\eta _ f$ and $\text{Tr}_ f$ this is Lemmas 86.6.2 and 86.6.1. For the arrow using Cohomology on Sites, Equation (21.50.0.1) this is Cohomology on Sites, Remark 21.50.2. For the multiplication map it is clear. This finishes the proof. $\square$
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