The Stacks project

Lemma 30.26.8. Consider a cartesian diagram of schemes

\[ \xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^ f \\ S' \ar[r]^ g & S } \]

with $f$ locally of finite type. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_ X$-module. If the support of $\mathcal{F}$ is proper over $S$, then the support of $(g')^*\mathcal{F}$ is proper over $S'$.

Proof. Observe that the statement makes sense because $(g')*\mathcal{F}$ is of finite type by Modules, Lemma 17.9.2. We have $\text{Supp}((g')^*\mathcal{F}) = (g')^{-1}(\text{Supp}(\mathcal{F}))$ by Morphisms, Lemma 29.5.3. Thus the lemma follows from Lemma 30.26.4. $\square$

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