# A scheme from a quasi-compact morphism

Let $f: X \to Y$ be quasi-compact, set $\mathcal{J} = \ker f^{\#}$, and $Z = V(\ker f^{\#}) = \{ y \in Y \: | \: \mathcal{J}_y \neq \mathcal{O}_{Y,y} \}$. $Z$ is a locally ringed topological space with sheaf $j^{-1}( \mathcal{O}_Y/ \mathcal{J})$, $j$ being the inclusion $Z \to Y$.

Why is $Z$ a scheme?

This is an exercise in Liu (2.3.17(c)) and the hint says that one should draw inspiration from the proofs of Propositions 3.12 and 3.20. However, I can't see how to begin.

This is a question from an assigned problem set. Please do not solve this for me. Instead I would greatly appreciate a nod in the right direction.

(For completeness, let me mention what the two propositions mentioned are. The first says that if $X$ is a scheme admitting a finite cover by affine opens $U_i$ such that each $U_i \cap U_j$ also admits such a covering, then restriction $\mathcal{O}_X(X) \to \mathcal{O}_X(X_f)$ induces an isomorphism $\mathcal{O}_X(X)_f \to \mathcal{O}_X(X_f)$, where $f \in \mathcal{O}_X(X)$ and $X_f = \{ x \in X \: | \: f \in \mathcal{O}_{X,x}^{\times} \}$. The second is the result characterising the closed immersions into an affine scheme.)

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Hint: Try to show that $J$ is quasi-coherent.
In my opinion this exercise is a waste of time without the knowledge of some basic notions. Of course you can solve it directly, but it gets cumbersome. Anyway, here is another hint: Reduce to the case that $Y$ is affine, say with ring $A$. Try to show that $Z$ is isomorphic to $\mathrm{Spec}(A/I)$, where $I$ is the ideal of global sections of $\mathcal{J}$. –  Martin Brandenburg Mar 30 at 16:54