Let $\mathcal K$ be a Grothendieck category. Recall the Gabriel filtration $0 \subseteq \mathcal K_1 \subseteq \cdots \mathcal K$ of localizing subcategories, where $\mathcal K_{\alpha+1}$ is generated by $\mathcal K_\alpha$ along with the finite-length objects of $\mathcal K^\alpha := \mathcal K / \mathcal K_\alpha$ (and if $\alpha$ is limit then $\mathcal K_\alpha$ is generated by the $\mathcal K_\beta$ for $\beta < \alpha$). If $\mathcal K_\alpha = \mathcal K$ for some ordinal $\alpha$, then $\mathcal K$ is said to be a Gabriel category and $\alpha$ is its Gabriel dimension.
In des categories abeliennes, Gabriel asserts without proof (Section IV, Prop 1 — note that he says “Krull dimension” instead of “Gabriel dimension”) that if $\mathcal K$ is a Gabriel category and $\mathcal T \subseteq \mathcal K$ is a localizing subcategory, then $\mathcal T$ and $\mathcal K / \mathcal T$ are both Gabriel categories.
Question: How does one prove this (that Gabriel categories are stable under localization)?
Gabriel says it follows from the definitions. But I’m at a loss: the definition of $\mathcal K_{\alpha+1}$ seems to me to be quite sensitive to the structure of $\mathcal K / \mathcal K_\alpha$, so I’m having difficulty relating the Gabriel filtration of $\mathcal K$ to the Gabriel filtration of $\mathcal K / \mathcal T$ or to that of $\mathcal T$.
As a further clue, Gabriel also asserts that $$max(dim(\mathcal T), dim(\mathcal K / \mathcal T)) \leq dim(\mathcal K) \leq dim(\mathcal T) + dim(\mathcal K / \mathcal T),$$ which really suggests the proof is supposed to do with some kind of elementary splicing together of Gabriel filtrations (especially since that’s an ordinal sum! See the notation at the beginning of Section IV.), but I am quite confused.
An analogous question one category level down would be whether hypoabelian groups are stable under quotients. The answer here is no: free groups are hypoabelian.
