Coherence of some sheaf in the proof of the Lemma de dévissage (Gortz's Algebraic Geometry).

Question

I'm reading Gortz's Algebraic Geometry, Lemma 12.63 and I don't understand some statement (Coherence of a sheaf in the proof ) :

Why the underlined statement is true? ; i.e., why $\mathcal{G}$ is coherent? ( If necessary, I will upload a detailed argument in the proof )

1. If $j_{*}\mathcal{H}|_{U}$ is coherent, then since image of homomorphism between coherent sheaf is coherent and (finite) direct sum of coherent sheaf is also coherent, we are done. But is it true? Note that $\mathcal{H}|_{U}$ is coherent since $j : U \to X$ is a morphism of noetherian schemes (Hartshorne, p.115, Prop.5.8). And pushforward $j_{*}\mathcal{H}|_{U}$ is also coherent?

2. If $j_{*}\mathcal{H}|_{U}$ is not coherent in general, nevertheless $\mathcal{G} = \operatorname{Im}(w) \oplus \operatorname{Im}(i)$ ( $i : \mathcal{H} \to j_{*}\mathcal{H}|_{U}$) is coherent? Note that first, $\operatorname{Im}(w)$ and $\operatorname{Im}(i)$ are quasi-coherent since each $\mathcal{F}$, $\mathcal{H}$, $j_{*}\mathcal{H}|_{U}$ are quasi-coherent (Direct image of quasi-coherent sheaf under a quasi-compact, quasi-separated morphism (e.g. open immersion from noetherian scheme) is also quasi-coherent). Second, on noetherian scheme, finite typeness together quasi-coherence implies coherence. So it suffices to show that $\operatorname{Im}(w)$ and $\operatorname{Im}(i)$ are finite type. And is it ture?

I'm struggling with this issue and I can't prove it completely at all now.

Can anyone help?

Answer 1

What I think is going on here is that, in your notation, the quasi-coherent sub-$\mathcal{O}_X$-module $\ker(i) \subseteq \mathcal{H}$ is also coherent, as $\mathcal{H}$ is. Thus the image $\text{im}(i) \cong \mathcal{H}/\ker(i)$ is coherent as coherent sheaves form an abelian subcategory.

In other words, I think (2) is the right statement.