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 on 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:
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).
The isomorphisms of (1) are compatible with pullbacks.
There is a canonical map
\pi ^*\mathcal{M}_0 \longrightarrow \widetilde{\mathcal{M}}.
The construction \mathcal{M} \mapsto \widetilde{\mathcal{M}} is functorial in \mathcal{M}.
The construction \mathcal{M} \mapsto \widetilde{\mathcal{M}} is exact.
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.
There exist canonical maps
\pi ^*\mathcal{M} \longrightarrow \bigoplus \nolimits _{n \in \mathbf{Z}} \widetilde{\mathcal{M}(n)}
generalizing (27.10.1.6).
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
Comments (3)
Comment #7509 by Matthieu Romagny on
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Comment #9967 by Santiago Arango on