Lemma 103.17.4 (0GR8)—The Stacks project

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Part 7: Algebraic Stacks
Chapter 103: Cohomology of Algebraic Stacks
Section 103.17: Coherent sheaves on locally Noetherian stacks
Lemma 103.17.4 (cite)

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Lemma 103.17.4. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of locally Noetherian algebraic stacks. Then $f^*$ sends coherent modules on $\mathcal{Y}$ to coherent modules on $\mathcal{X}$ .

Proof. Immediate from the definition and the fact that pullback for any morphism of ringed topoi preserves finitely presented modules, see Modules on Sites, Lemma 18.23.4. $\square$

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