The Stacks project

Lemma 70.7.3. With notation and assumptions as in Lemma 70.7.1. Then

  1. any finite locally free $\mathcal{O}_ X$-module is the pullback of a finite locally free $\mathcal{O}_{X_ i}$-module for some $i$,

  2. any invertible $\mathcal{O}_ X$-module is the pullback of an invertible $\mathcal{O}_{X_ i}$-module for some $i$.

Proof. Proof of (2). Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Since invertible modules are of finite presentation we can find an $i$ and modules $\mathcal{L}_ i$ and $\mathcal{N}_ i$ of finite presentation over $X_ i$ such that $f_ i^*\mathcal{L}_ i \cong \mathcal{L}$ and $f_ i^*\mathcal{N}_ i \cong \mathcal{L}^{\otimes -1}$, see Lemma 70.7.2. Since pullback commutes with tensor product we see that $f_ i^*(\mathcal{L}_ i \otimes _{\mathcal{O}_{X_ i}} \mathcal{N}_ i)$ is isomorphic to $\mathcal{O}_ X$. Since the tensor product of finitely presented modules is finitely presented, the same lemma implies that $f_{i'i}^*\mathcal{L}_ i \otimes _{\mathcal{O}_{X_{i'}}} f_{i'i}^*\mathcal{N}_ i$ is isomorphic to $\mathcal{O}_{X_{i'}}$ for some $i' \geq i$. It follows that $f_{i'i}^*\mathcal{L}_ i$ is invertible (Modules on Sites, Lemma 18.32.2) and the proof is complete.

Proof of (1). Omitted. Hint: argue as in the proof of (2) using that a module (on a locally ringed site) is finite locally free if and only if it has a dual, see Modules on Sites, Section 18.29. Alternatively, argue as in the proof for schemes, see Limits, Lemma 32.10.3. $\square$

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