The Stacks project

Lemma 12.19.4. Let $\mathcal{A}$ be an abelian category. Let $f : A \to B$ be a morphism of filtered objects of $\mathcal{A}$. The following are equivalent

  1. $f$ is strict,

  2. the morphism $\mathop{\mathrm{Coim}}(f) \to \mathop{\mathrm{Im}}(f)$ of Lemma 12.3.12 is an isomorphism.

Proof. Note that $\mathop{\mathrm{Coim}}(f) \to \mathop{\mathrm{Im}}(f)$ is an isomorphism of objects of $\mathcal{A}$, and that part (2) signifies that it is an isomorphism of filtered objects. By the description of kernels and cokernels in the proof of Lemma 12.19.2 we see that the filtration on $\mathop{\mathrm{Coim}}(f)$ is the quotient filtration coming from $A \to \mathop{\mathrm{Coim}}(f)$. Similarly, the filtration on $\mathop{\mathrm{Im}}(f)$ is the induced filtration coming from the injection $\mathop{\mathrm{Im}}(f) \to B$. The definition of strict is exactly that the quotient filtration is the induced filtration. $\square$

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