The Stacks project

84.7 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 84.3.1. 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 73.20.1) for which we use the trace map $Rf_*a(L) \to L$. When $L = \mathcal{O}_ Y$ we obtain a map
\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.

Lemma 84.7.1. 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, ( is an isomorphism if $K$ is perfect.

Proof. Let $K^\vee $ be the “dual” to $K$, see Cohomology on Sites, Lemma 21.46.4. 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 21.48.1) or the more general Derived Categories of Spaces, Lemma 73.20.1. Hence the result by the Yoneda lemma. $\square$

Lemma 84.7.2. Suppose we have a diagram ( Let $K \in D_\mathit{QCoh}(\mathcal{O}_ Y)$. The diagram

\[ \xymatrix{ L(g')^*(Lf^*K \otimes ^\mathbf {L}_{\mathcal{O}_ X} a(\mathcal{O}_ Y)) \ar[r] \ar[d] & L(g')^*a(K) \ar[d] \\ L(f')^*Lg^*K \otimes _{\mathcal{O}_{X'}}^\mathbf {L} a'(\mathcal{O}_{Y'}) \ar[r] & a'(Lg^*K) } \]

commutes where the horizontal arrows are the maps ( for $K$ and $Lg^*K$ and the vertical maps are constructed using Cohomology on Sites, Remark 21.19.3 and (

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 ( 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 ( which is an isomorphism by Derived Categories of Spaces, Lemma 73.20.1, 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 84.6.2 and 84.6.1. For the arrow using Cohomology on Sites, Equation ( this is Cohomology on Sites, Remark 21.48.2. For the multiplication map it is clear. This finishes the proof. $\square$

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