Section 20.54 (01E6): Projection formula

20.54 Projection formula

In this section we collect variants of the projection formula. The most basic version is Lemma 20.54.2. After we state and prove it, we discuss a more general version involving perfect complexes.

Lemma 20.54.1. Let $X$ be a ringed space. Let $\mathcal{I}$ be an injective $\mathcal{O}_ X$ -module. Let $\mathcal{E}$ be an $\mathcal{O}_ X$ -module. Assume $\mathcal{E}$ is finite locally free on $X$ , see Modules, Definition 17.14.1. Then $\mathcal{E} \otimes _{\mathcal{O}_ X} \mathcal{I}$ is an injective $\mathcal{O}_ X$ -module.

Proof. This is true because under the assumptions of the lemma we have

$\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{E} \otimes _{\mathcal{O}_ X} \mathcal{I}) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}( \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{E}^\vee , \mathcal{I})$

where $\mathcal{E}^\vee = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}, \mathcal{O}_ X)$ is the dual of $\mathcal{E}$ which is finite locally free also. Since tensoring with a finite locally free sheaf is an exact functor we win by Homology, Lemma 12.27.2. $\square$

Lemma 20.54.2. Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{F}$ be an $\mathcal{O}_ X$ -module. Let $\mathcal{E}$ be an $\mathcal{O}_ Y$ -module. Assume $\mathcal{E}$ is finite locally free on $Y$ , see Modules, Definition 17.14.1. Then there exist isomorphisms

$\mathcal{E} \otimes _{\mathcal{O}_ Y} R^ qf_*\mathcal{F} \longrightarrow R^ qf_*(f^*\mathcal{E} \otimes _{\mathcal{O}_ X} \mathcal{F})$

for all $q \geq 0$ . In fact there exists an isomorphism

$\mathcal{E} \otimes _{\mathcal{O}_ Y} Rf_*\mathcal{F} \longrightarrow Rf_*(f^*\mathcal{E} \otimes _{\mathcal{O}_ X} \mathcal{F})$

in $D^{+}(Y)$ functorial in $\mathcal{F}$ .

Proof. Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$ on $X$ . Note that $f^*\mathcal{E}$ is finite locally free also, hence we get a resolution

$f^*\mathcal{E} \otimes _{\mathcal{O}_ X} \mathcal{F} \longrightarrow f^*\mathcal{E} \otimes _{\mathcal{O}_ X} \mathcal{I}^\bullet$

which is an injective resolution by Lemma 20.54.1. Apply $f_*$ to see that

$Rf_*(f^*\mathcal{E} \otimes _{\mathcal{O}_ X} \mathcal{F}) = f_*(f^*\mathcal{E} \otimes _{\mathcal{O}_ X} \mathcal{I}^\bullet ).$

Hence the lemma follows if we can show that $f_*(f^*\mathcal{E} \otimes _{\mathcal{O}_ X} \mathcal{F}) = \mathcal{E} \otimes _{\mathcal{O}_ Y} f_*(\mathcal{F})$ functorially in the $\mathcal{O}_ X$ -module $\mathcal{F}$ . This is clear when $\mathcal{E} = \mathcal{O}_ Y^{\oplus n}$ , and follows in general by working locally on $Y$ . Details omitted. $\square$

Let $f : X \to Y$ be a morphism of ringed spaces. Let $E \in D(\mathcal{O}_ X)$ and $K \in D(\mathcal{O}_ Y)$ . Without any further assumptions there is a map

20.54.2.1

$\begin{equation} \label{cohomology-equation-projection-formula-map} Rf_*E \otimes ^\mathbf {L}_{\mathcal{O}_ Y} K \longrightarrow Rf_*(E \otimes ^\mathbf {L}_{\mathcal{O}_ X} Lf^*K) \end{equation}$

Namely, it is the adjoint to the canonical map

$Lf^*(Rf_*E \otimes ^\mathbf {L}_{\mathcal{O}_ Y} K) = Lf^*Rf_*E \otimes ^\mathbf {L}_{\mathcal{O}_ X} Lf^*K \longrightarrow E \otimes ^\mathbf {L}_{\mathcal{O}_ X} Lf^*K$

coming from the map $Lf^*Rf_*E \to E$ and Lemmas 20.27.3 and 20.28.1. A reasonably general version of the projection formula is the following.

Lemma 20.54.3. Let $f : X \to Y$ be a morphism of ringed spaces. Let $E \in D(\mathcal{O}_ X)$ and $K \in D(\mathcal{O}_ Y)$ . If $K$ is perfect, then

$Rf_*E \otimes ^\mathbf {L}_{\mathcal{O}_ Y} K = Rf_*(E \otimes ^\mathbf {L}_{\mathcal{O}_ X} Lf^*K)$

in $D(\mathcal{O}_ Y)$ .

Proof. To check (20.54.2.1) is an isomorphism we may work locally on $Y$ , i.e., we have to find an open covering $\{ V_ j \to Y\}$ such that the map restricts to an isomorphism on $V_ j$ . By definition of perfect objects, this means we may assume $K$ is represented by a strictly perfect complex of $\mathcal{O}_ Y$ -modules. Note that, completely generally, the statement is true for $K = K_1 \oplus K_2$ , if and only if the statement is true for $K_1$ and $K_2$ . Hence we may assume $K$ is a finite complex of finite free $\mathcal{O}_ Y$ -modules. In this case a simple argument involving stupid truncations reduces the statement to the case where $K$ is represented by a finite free $\mathcal{O}_ Y$ -module. Since the statement is invariant under finite direct summands in the $K$ variable, we conclude it suffices to prove it for $K = \mathcal{O}_ Y[n]$ in which case it is trivial. $\square$

Here is a case where the projection formula is true in complete generality.

Lemma 20.54.4. Let $f : X \to Y$ be a morphism of ringed spaces such that $f$ is a homeomorphism onto a closed subset. Then (20.54.2.1) is an isomorphism always.

Proof. Since $f$ is a homeomorphism onto a closed subset, the functor $f_*$ is exact (Modules, Lemma 17.6.1). Hence $Rf_*$ is computed by applying $f_*$ to any representative complex. Choose a K-flat complex $\mathcal{K}^\bullet$ of $\mathcal{O}_ Y$ -modules representing $K$ and choose any complex $\mathcal{E}^\bullet$ of $\mathcal{O}_ X$ -modules representing $E$ . Then $Lf^*K$ is represented by $f^*\mathcal{K}^\bullet$ which is a K-flat complex of $\mathcal{O}_ X$ -modules (Lemma 20.26.8). Thus the right hand side of (20.54.2.1) is represented by

$f_*\text{Tot}(\mathcal{E}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{K}^\bullet )$

By the same reasoning we see that the left hand side is represented by

$\text{Tot}(f_*\mathcal{E}^\bullet \otimes _{\mathcal{O}_ Y} \mathcal{K}^\bullet )$

Since $f_*$ commutes with direct sums (Modules, Lemma 17.6.3) it suffices to show that

$f_*(\mathcal{E} \otimes _{\mathcal{O}_ X} f^*\mathcal{K}) = f_*\mathcal{E} \otimes _{\mathcal{O}_ Y} \mathcal{K}$

for any $\mathcal{O}_ X$ -module $\mathcal{E}$ and $\mathcal{O}_ Y$ -module $\mathcal{K}$ . We will check this by checking on stalks. Let $y \in Y$ . If $y \not\in f(X)$ , then the stalks of both sides are zero. If $y = f(x)$ , then we see that we have to show

$\mathcal{E}_ x \otimes _{\mathcal{O}_{X, x}} (\mathcal{O}_{X, x} \otimes _{\mathcal{O}_{Y, y}} \mathcal{F}_ y) = \mathcal{E}_ x \otimes _{\mathcal{O}_{Y, y}} \mathcal{F}_ y$

(using Sheaves, Lemma 6.32.1 and Lemma 6.26.4). This equality holds and therefore the lemma has been proved. $\square$

Remark 20.54.5. The map (20.54.2.1) is compatible with the base change map of Remark 20.28.3 in the following sense. Namely, suppose that

$\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y }$

is a commutative diagram of ringed spaces. Let $E \in D(\mathcal{O}_ X)$ and $K \in D(\mathcal{O}_ Y)$ . Then the diagram

$\xymatrix{ Lg^*(Rf_*E \otimes ^\mathbf {L}_{\mathcal{O}_ Y} K) \ar[r]_ p \ar[d]_ t & Lg^*Rf_*(E \otimes ^\mathbf {L}_{\mathcal{O}_ X} Lf^*K) \ar[d]_ b \\ Lg^*Rf_*E \otimes ^\mathbf {L}_{\mathcal{O}_{Y'}} Lg^*K \ar[d]_ b & Rf'_*L(g')^*(E \otimes ^\mathbf {L}_{\mathcal{O}_ X} Lf^*K) \ar[d]_ t \\ Rf'_*L(g')^*E \otimes ^\mathbf {L}_{\mathcal{O}_{Y'}} Lg^*K \ar[rd]_ p & Rf'_*(L(g')^*E \otimes ^\mathbf {L}_{\mathcal{O}_{Y'}} L(g')^*Lf^*K) \ar[d]_ c \\ & Rf'_*(L(g')^*E \otimes ^\mathbf {L}_{\mathcal{O}_{Y'}} L(f')^*Lg^*K) }$

is commutative. Here arrows labeled $t$ are gotten by an application of Lemma 20.27.3, arrows labeled $b$ by an application of Remark 20.28.3, arrows labeled $p$ by an application of (20.54.2.1), and $c$ comes from $L(g')^* \circ Lf^* = L(f')^* \circ Lg^*$ . We omit the verification.

The Stacks project

20.54 Projection formula

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