The Stacks project

27.17 Quasi-coherent sheaves on relative Proj

We briefly discuss how to deal with graded modules in the relative setting.

We place ourselves in Situation 27.15.1. So $S$ is a scheme, and $\mathcal{A}$ is a quasi-coherent graded $\mathcal{O}_ S$-algebra. Let $\mathcal{M} = \bigoplus _{n \in \mathbf{Z}} \mathcal{M}_ n$ be a graded $\mathcal{A}$-module, quasi-coherent as an $\mathcal{O}_ S$-module. We are going to describe the associated quasi-coherent sheaf of modules on $\underline{\text{Proj}}_ S(\mathcal{A})$. We first describe the value of this sheaf schemes $T$ mapping into the relative Proj.

Let $T$ be a scheme. Let $(d, f : T \to S, \mathcal{L}, \psi )$ be a quadruple over $T$, as in Section 27.16. We define a quasi-coherent sheaf $\widetilde{\mathcal{M}}_ T$ of $\mathcal{O}_ T$-modules as follows

27.17.0.1
\begin{equation} \label{constructions-equation-widetilde-M} \widetilde{\mathcal{M}}_ T = \left( f^*\mathcal{M}^{(d)} \otimes _{f^*\mathcal{A}^{(d)}} \left(\bigoplus \nolimits _{n \in \mathbf{Z}} \mathcal{L}^{\otimes n}\right) \right)_0 \end{equation}

So $\widetilde{\mathcal{M}}_ T$ is the degree $0$ part of the tensor product of the graded $f^*\mathcal{A}^{(d)}$-modules $\mathcal{M}^{(d)}$ and $\bigoplus \nolimits _{n \in \mathbf{Z}} \mathcal{L}^{\otimes n}$. Note that the sheaf $\widetilde{\mathcal{M}}_ T$ depends on the quadruple even though we suppressed this in the notation. This construction has the pleasing property that given any morphism $g : T' \to T$ we have $\widetilde{\mathcal{M}}_{T'} = g^*\widetilde{\mathcal{M}}_ T$ where $\widetilde{\mathcal{M}}_{T'}$ denotes the quasi-coherent sheaf associated to the pullback quadruple $(d, f \circ g, g^*\mathcal{L}, g^*\psi )$.

Since all sheaves in (27.17.0.1) are quasi-coherent we can spell out the construction over an affine open $\mathop{\mathrm{Spec}}(C) = V \subset T$ which maps into an affine open $\mathop{\mathrm{Spec}}(R) = U \subset S$. Namely, suppose that $\mathcal{A}|_ U$ corresponds to the graded $R$-algebra $A$, that $\mathcal{M}|_ U$ corresponds to the graded $A$-module $M$, and that $\mathcal{L}|_ V$ corresponds to the invertible $C$-module $L$. The map $\psi $ gives rise to a graded $R$-algebra map $\gamma : A^{(d)} \to \bigoplus _{n \geq 0} L^{\otimes n}$. (Tensor powers of $L$ over $C$.) Then $(\widetilde{\mathcal{M}}_ T)|_ V$ is the quasi-coherent sheaf associated to the $C$-module

\[ N_{R, C, A, M, \gamma } = \left( M^{(d)} \otimes _{A^{(d)}, \gamma } \left(\bigoplus \nolimits _{n \in \mathbf{Z}} L^{\otimes n}\right) \right)_0 \]

By assumption we may even cover $T$ by affine opens $V$ such that there exists some $a \in A_ d$ such that $\gamma (a) \in L$ is a $C$-basis for the module $L$. In that case any element of $N_{R, C, A, M, \gamma }$ is a sum of pure tensors $\sum m_ i \otimes \gamma (a)^{-n_ i}$ with $m \in M_{n_ id}$. In fact we may multiply each $m_ i$ with a suitable positive power of $a$ and collect terms to see that each element of $N_{R, C, A, M, \gamma }$ can be written as $m \otimes \gamma (a)^{-n}$ with $m \in M_{nd}$ and $n \gg 0$. In other words we see that in this case

\[ N_{R, C, A, M, \gamma } = M_{(a)} \otimes _{A_{(a)}} C \]

where the map $A_{(a)} \to C$ is the map $x/a^ n \mapsto \gamma (x)/\gamma (a)^ n$. In other words, this is the value of $\widetilde{M}$ on $D_{+}(a) \subset \text{Proj}(A)$ pulled back to $\mathop{\mathrm{Spec}}(C)$ via the morphism $\mathop{\mathrm{Spec}}(C) \to D_{+}(a)$ coming from $\gamma $.

Lemma 27.17.1. In Situation 27.15.1. For any quasi-coherent sheaf of graded $\mathcal{A}$-modules $\mathcal{M}$ on $S$, there exists a canonical associated sheaf of $\mathcal{O}_{\underline{\text{Proj}}_ S(\mathcal{A})}$-modules $\widetilde{\mathcal{M}}$ with the following properties:

  1. Given a scheme $T$ and a quadruple $(T \to S, d, \mathcal{L}, \psi )$ over $T$ corresponding to a morphism $h : T \to \underline{\text{Proj}}_ S(\mathcal{A})$ there is a canonical isomorphism $\widetilde{\mathcal{M}}_ T = h^*\widetilde{\mathcal{M}}$ where $\widetilde{\mathcal{M}}_ T$ is defined by (27.17.0.1).

  2. The isomorphisms of (1) are compatible with pullbacks.

  3. There is a canonical map

    \[ \pi ^*\mathcal{M}_0 \longrightarrow \widetilde{\mathcal{M}}. \]
  4. The construction $\mathcal{M} \mapsto \widetilde{\mathcal{M}}$ is functorial in $\mathcal{M}$.

  5. The construction $\mathcal{M} \mapsto \widetilde{\mathcal{M}}$ is exact.

  6. There are canonical maps

    \[ \widetilde{\mathcal{M}} \otimes _{\mathcal{O}_{\underline{\text{Proj}}_ S(\mathcal{A})}} \widetilde{\mathcal{N}} \longrightarrow \widetilde{\mathcal{M} \otimes _\mathcal {A} \mathcal{N}} \]

    as in Lemma 27.9.1.

  7. There exist canonical maps

    \[ \pi ^*\mathcal{M} \longrightarrow \bigoplus \nolimits _{n \in \mathbf{Z}} \widetilde{\mathcal{M}(n)} \]

    generalizing (27.10.1.6).

  8. The formation of $\widetilde{\mathcal{M}}$ commutes with base change.

Proof. Omitted. We should split this lemma into parts and prove the parts separately. $\square$


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