Lemma 78.13.3. Let S be a scheme. Let f : Y \to X be a morphism of algebraic spaces over S. Let \mathcal{F} be a quasi-coherent \mathcal{O}_ X-module, let \mathcal{G} be a quasi-coherent \mathcal{O}_ Y-module, and let \varphi : \mathcal{G} \to f^*\mathcal{F} be a module map. Assume
\varphi is injective,
f is quasi-compact, quasi-separated, flat, and surjective,
X, Y are locally Noetherian, and
\mathcal{G} is a coherent \mathcal{O}_ Y-module.
Then \mathcal{F} \cap f_*\mathcal{G} defined as the pullback
\xymatrix{ \mathcal{F} \ar[r] & f_*f^*\mathcal{F} \\ \mathcal{F} \cap f_*\mathcal{G} \ar[u] \ar[r] & f_*\mathcal{G} \ar[u] }
is a coherent \mathcal{O}_ X-module.
Proof.
We will freely use the characterization of coherent modules of Cohomology of Spaces, Lemma 69.12.2 as well as the fact that coherent modules form a Serre subcategory of \mathit{QCoh}(\mathcal{O}_ X), see Cohomology of Spaces, Lemma 69.12.4. If f has a section \sigma , then we see that \mathcal{F} \cap f_*\mathcal{G} is contained in the image of \sigma ^*\mathcal{G} \to \sigma ^*f^*\mathcal{F} = \mathcal{F}, hence coherent. In general, to show that \mathcal{F} \cap f_*\mathcal{G} is coherent, it suffices the show that f^*(\mathcal{F} \cap f_*\mathcal{G}) is coherent (see Descent on Spaces, Lemma 74.6.1). Since f is flat this is equal to f^*\mathcal{F} \cap f^*f_*\mathcal{G}. Since f is flat, quasi-compact, and quasi-separated we see f^*f_*\mathcal{G} = p_*q^*\mathcal{G} where p, q : Y \times _ X Y \to Y are the projections, see Cohomology of Spaces, Lemma 69.11.2. Since p has a section we win.
\square
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