I'm trying to understand a sufficient condition for the existence of a quasi-compact base of open sets of Zariski topology on the spectrum of an abelian category from https://sasharosenberg.com/?x-portfolio=noncommutative-local-algebra page 29.
Consider the proposition: Let $\mathcal{A}$ be an abelian category with property (Sup) and a generator of finite type. Let $\Omega$ be a family of closed subcategories of the category $\mathcal{A}$ such that the intersection $\mathcal{S}$ of all the subcategories from $\Omega$ is finite.Then $\mathcal{S}$ is the intersection of finite number of categories from $\Omega$.
Now I understand the proof of this proposition. But I do not understand how it implies the following proposition: Let $\mathcal{A}$ be an abelian category with property (Sup) and a generator of finite type. Then the Zariski topology is quasicompact.
To be precise, the base of closed sets of the Zariski topology is defined by taking the spectrum over left closed sub-categories of $\mathcal{A}.$ Then how does one identify closed sub-categories in this topology so that one is able to use the first proposition? Sure, I can look into the spectrum of $\mathcal{A^{op}}$, where a left closed subcategory of $\mathcal{A}$ becomes closed subcategory. But then we may not have $\mathcal{A^{op}}$ with (Sup) and a generator of finite type.
Thanks in advance!