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

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?

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.
• The coherence of $\operatorname{im}(w)$ can be showed by similar argument? Nov 2, 2022 at 13:06