Let X scheme, $\rho \subset \mathcal{O}_X$ a quasicoherent ideal sheaf. I want prove that $Supp(\mathcal{O}_X / \rho) := \{a \in X | (\mathcal{O}_X / \rho)_a \neq 0\}$ is closed. I already proved it for $X$ affine. Is it posible to reduce the statement to the alredy proved affine case?
My problem is that if I cover $X$ by open affine sets $U_i$ such that $X = \bigcup _i U_i$ and want to use the already proven affine case that $Supp(\mathcal{O}_X / \rho) \cap U_i$ closed in $U_i$, I can't deduce from this that $Supp(\mathcal{O}_X / \rho)$ is in $X$ closed because $U_i$ are open. Is there any posibility to "save" this argument?
Maybe by using cover property?
Yes, it actually does suffice to check closedness of a set on an open cover of the space. In general, suppose $X$ is a topological space, $\{U_i\}$ is an open cover, and $C\subseteq X$ is such that $C\cap U_i$ is closed in $U_i$ for all $i$. Then $U_i\setminus C$ is open in $U_i$, and hence open in $X$ since $U_i$ is open in $X$. Thus $$X\setminus C=\bigcup_i (X\setminus C)\cap U_i=\bigcup_i U_i\setminus C$$ is a union of open sets and hence open, so $C$ is closed.