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

84.7.0.1
\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.

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, (84.7.0.1) 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 (84.4.0.1). 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 (84.7.0.1) for $K$ and $Lg^*K$ and the vertical maps are constructed using Cohomology on Sites, Remark 21.19.3 and (84.4.1.1).

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 (84.7.0.1) 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 (21.48.0.1) 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 (21.48.0.1) 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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