Let $X$ be a quasicompact, quasiseparated scheme and $Z$ a closed subset of $X$ with quasicompact complement. Is there a finite type quasicoherent ideal sheaf $\mathcal{J}$ on $X$ which cuts out $Z$? (more formally, such that $\mathcal{O}_X/\mathcal{J}$ has support $Z$?)
In the affine case the answer is yes: $V(\mathfrak{a})$ has quasicompact complement iff $\mathfrak{a}$ is finitely generated up to radical. But I'm having trouble globalizing this. We can take a finite cover of $X$ by affines (and cover all pairwise intersections by finitely many mutually distinguished opens) and take arbitrary finitely generated defining ideals on each affine, but I don't see how we can modify these such that they glue together to a global ideal sheaf.
Here is what I've thought about so far: write $X = U_1 \cup \ldots \cup U_n$ for $U_i$ affine and $U_{ij} = V_{ij}^1 \cup \ldots \cup V_{ij}^{k_{ij}}$. Let $\mathfrak{a}_i \leq \mathcal{O}_X(U_i)$ be a finitely generated ideal with $V(\mathfrak{a}_i) = Z \cap U_i$. We can choose $N$ uniform such that for any $i, j$ we have $(\mathfrak{a}_i \mathcal{O}_X(U_{ij}))^N \subseteq \mathfrak{a}_j \mathcal{O}_X(U_{ij})$. In fact more is true: write $U_{ij} = D(f_i)$ for $f_i \in \mathcal{O}_X(U_i)$ and $\mathfrak{a}_i = (a_{i,1},\ldots,a_{i, r_i})$; then there is some uniform $K$ such that for any $i, j$ and any product of $N$ generators of $ \mathfrak{a}_i$ there is a relation showing that that product lies in $\mathfrak{a}_j \mathcal{O}_X(U_{ij})$ whose coefficients' denominators have power of $f_j$ at most $K$. But I don't see how any of this actually helps.
I would prefer a "global" or "all at once" answer rather than one which uses the induction principle for qcqs schemes.
Edit: Here is a construction which I think works, but I'm unsure. Cover $X$ by finitely many affine opens and cover the pairwise intersections by finitely many simultaneously distinguished opens. Further iterated intersections will be affine (they're distinguished in some cover element) and there are only finitely many of them overall. So we have a finite set $\mathcal{P}$ of affine opens which cover $X$ and where the pairwise intersection of any two elements is covered by elements. Since $\mathcal{P}$ is finite it's in particular well founded and so we can define functions on it by recursion; we use this to define a family of finitely generated ideals $I(U) \leq \mathcal{O}_X(U)$ such that $I(U)$ has vanishing locus $Z \cap U$ and $I(U) \mathcal{O}_X(V) \subseteq I(V)$ whenever $V\subseteq U$. Let $U\in \mathcal{P}$ be arbitrary and suppose $I(V)$ has been defined for all $V \subsetneq U$ in $\mathcal{P}$. Because $U$ is affine and $Z\cap U$ is a closed subset with quasicompact complement there is a finitely generated ideal $J \leq \mathcal{O}_X(U)$ with vanishing locus $Z\cap U$. Then for each $V \subsetneq U$ the ideal $J \mathcal{O}_X(V)$ is finitely generated and contained in $\sqrt{I(V)}$, which implies there is some $n$ such that $J^n \mathcal{O}_X(V) \leq I(V)$. Since there are only finitely many $V$ we can find a uniform $N$ such that for any $V$ we have $J^N \mathcal{O}_X(V) \leq I(V)$. Define $I(U) = J^N$.
This construction gives us a diagram of ideals of $\mathcal{O}_X(U)$ for $U\in \mathcal{P}$. If we let $i_U : U \to X$ be the inclusion then we can also view $U \mapsto (i_U)_*i_U^*\widetilde{I(U)}$ as a functor $\mathcal{P}^{\mathrm{op}} \to \mathcal{O}_X-\mathsf{Mod}$ equipped with a monomorphism to the functor $U \mapsto (i_U)_*i_U^*\mathcal{O}_X$. Since $X$ is qcqs all these pushforwards are in fact quasicoherent modules, and the sheaf condition tells us the limit of that latter functor is $\mathcal{O}_X$. Then $\mathcal{J} = \lim_{U\in\mathcal{P}} (i_U)_*i_U^*\widetilde{I(U)}$ is a quasicoherent sheaf equipped with a monomorphism to $\mathcal{O}_X$ (the limit of a family of monomorphisms between two diagrams is a monomorphism) ie is a quasicoherent ideal sheaf.
We built this $\mathcal{J}$ by locally constructing a fg ideal cutting out the right subset, so it feels like it should still be fg with the right support. But maybe taking this limit destroys any hope of finiteness? I guess the hope is that a finite limit of (sheaves of) rings equipped with finite type ideal sheaves is still finite type?