Lemma 42.23.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. The categories $\textit{Coh}_{\leq k}(X)$ are Serre subcategories of the abelian category $\textit{Coh}(X)$.

**Proof.**
The definition of a Serre subcategory is Homology, Definition 12.10.1. The proof of the lemma is straightforward and omitted.
$\square$

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