The Stacks project

Lemma 39.15.3. Let $f : Y \to X$ be a morphism of schemes. 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

  1. $\varphi $ is injective,

  2. $f$ is quasi-compact, quasi-separated, flat, and surjective,

  3. $X$, $Y$ are locally Noetherian, and

  4. $\mathcal{G}$ 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 Schemes, Lemma 30.9.1 as well as the fact that coherent modules form a Serre subcategory of $\mathit{QCoh}(\mathcal{O}_ X)$, see Cohomology of Schemes, Lemma 30.9.3. 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, Lemma 35.7.1). 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 Schemes, Lemma 30.5.2. Since $p$ has a section we win. $\square$

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