The Stacks project

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.