The Stacks project

Lemma 30.8.4. Let $S$ be a scheme. Let $n \geq 1$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ S$-module of constant rank $n + 1$. Consider the structure morphism

\[ \pi : \mathbf{P}(\mathcal{E}) \longrightarrow S. \]

We have

\[ R^ q\pi _*(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(d)) = \left\{ \begin{matrix} \text{Sym}^ d(\mathcal{E}) & \text{if} & q = 0 \\ 0 & \text{if} & q \not= 0, n \\ \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ S}( \text{Sym}^{- n - 1 - d}(\mathcal{E}) \otimes _{\mathcal{O}_ S} \wedge ^{n + 1}\mathcal{E}, \mathcal{O}_ S) & \text{if} & q = n \end{matrix} \right. \]

These identifications are compatible with base change and isomorphism between locally free sheaves.

Proof. Consider the canonical map

\[ \pi ^*\mathcal{E} \longrightarrow \mathcal{O}_{\mathbf{P}(\mathcal{E})}(1) \]

and twist down by $1$ to get

\[ \pi ^*(\mathcal{E})(-1) \longrightarrow \mathcal{O}_{\mathbf{P}(\mathcal{E})} \]

This is a surjective map from a locally free rank $n + 1$ sheaf onto the structure sheaf. Hence the corresponding Koszul complex is exact (More on Algebra, Lemma 15.28.5). In other words there is an exact complex

\[ 0 \to \pi ^*(\wedge ^{n + 1}\mathcal{E})(-n - 1) \to \ldots \to \pi ^*(\wedge ^ i\mathcal{E})(-i) \to \ldots \to \pi ^*\mathcal{E}(-1) \to \mathcal{O}_{\mathbf{P}(\mathcal{E})} \to 0 \]

We will think of the term $\pi ^*(\wedge ^ i\mathcal{E})(-i)$ as being in degree $-i$. We are going to compute the higher direct images of this acyclic complex using the first spectral sequence of Derived Categories, Lemma 13.21.3. Namely, we see that there is a spectral sequence with terms

\[ E_1^{p, q} = R^ q\pi _*\left(\pi ^*(\wedge ^{-p}\mathcal{E})(p)\right) \]

converging to zero! By the projection formula (Cohomology, Lemma 20.51.2) we have

\[ E_1^{p, q} = \wedge ^{-p} \mathcal{E} \otimes _{\mathcal{O}_ S} R^ q\pi _*\left(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(p)\right). \]

Note that locally on $S$ the sheaf $\mathcal{E}$ is trivial, i.e., isomorphic to $\mathcal{O}_ S^{\oplus n + 1}$, hence locally on $S$ the morphism $\mathbf{P}(\mathcal{E}) \to S$ can be identified with $\mathbf{P}^ n_ S \to S$. Hence locally on $S$ we can use the result of Lemmas 30.8.1, 30.8.2, or 30.8.3. It follows that $E_1^{p, q} = 0$ unless $(p, q)$ is $(0, 0)$ or $(-n - 1, n)$. The nonzero terms are

\begin{align*} E_1^{0, 0} & = \pi _*\mathcal{O}_{\mathbf{P}(\mathcal{E})} = \mathcal{O}_ S \\ E_1^{-n - 1, n} & = R^ n\pi _*\left(\pi ^*(\wedge ^{n + 1}\mathcal{E})(-n - 1)\right) = \wedge ^{n + 1}\mathcal{E} \otimes _{\mathcal{O}_ S} R^ n\pi _*\left(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(-n - 1)\right) \end{align*}

Hence there can only be one nonzero differential in the spectral sequence namely the map $d_{n + 1}^{-n - 1, n} : E_{n + 1}^{-n - 1, n} \to E_{n + 1}^{0, 0}$ which has to be an isomorphism (because the spectral sequence converges to the $0$ sheaf). Thus $E_1^{p, q} = E_{n + 1}^{p, q}$ and we obtain a canonical isomorphism

\[ \wedge ^{n + 1}\mathcal{E} \otimes _{\mathcal{O}_ S} R^ n\pi _*\left(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(-n - 1)\right) = R^ n\pi _*\left(\pi ^*(\wedge ^{n + 1}\mathcal{E})(-n - 1)\right) \xrightarrow {d_{n + 1}^{-n - 1, n}} \mathcal{O}_ S \]

Since $\wedge ^{n + 1}\mathcal{E}$ is an invertible sheaf, this implies that $R^ n\pi _*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(-n - 1)$ is invertible as well and canonically isomorphic to the inverse of $\wedge ^{n + 1}\mathcal{E}$. In other words we have proved the case $d = - n - 1$ of the lemma.

Working locally on $S$ we see immediately from the computation of cohomology in Lemmas 30.8.1, 30.8.2, or 30.8.3 the statements on vanishing of the lemma. Moreover the result on $R^0\pi _*$ is clear as well, since there are canonical maps $\text{Sym}^ d(\mathcal{E}) \to \pi _* \mathcal{O}_{\mathbf{P}(\mathcal{E})}(d)$ for all $d$. It remains to show that the description of $R^ n\pi _*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(d)$ is correct for $d < -n - 1$. In order to do this we consider the map

\[ \pi ^*(\text{Sym}^{-d - n - 1}(\mathcal{E})) \otimes _{\mathcal{O}_{\mathbf{P}(\mathcal{E})}} \mathcal{O}_{\mathbf{P}(\mathcal{E})}(d) \longrightarrow \mathcal{O}_{\mathbf{P}(\mathcal{E})}(-n - 1) \]

Applying $R^ n\pi _*$ and the projection formula (see above) we get a map

\[ \text{Sym}^{-d - n - 1}(\mathcal{E}) \otimes _{\mathcal{O}_ S} R^ n\pi _*(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(d)) \longrightarrow R^ n\pi _*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(-n - 1) = (\wedge ^{n + 1}\mathcal{E})^{\otimes -1} \]

(the last equality we have shown above). Again by the local calculations of Lemmas 30.8.1, 30.8.2, or 30.8.3 it follows that this map induces a perfect pairing between $R^ n\pi _*(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(d))$ and $\text{Sym}^{-d - n - 1}(\mathcal{E}) \otimes \wedge ^{n + 1}(\mathcal{E})$ as desired. $\square$


Comments (2)

Comment #940 by correction_bot on

  1. In the Koszul resolution, the term should be

  2. "In other to do this…", change other to order. Also, following this there are three occurrences of where is intended.


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