The Stacks project

Lemma 12.19.2. Let $\mathcal{A}$ be an abelian category. The category of filtered objects $\text{Fil}(\mathcal{A})$ has the following properties:

  1. It is an additive category.

  2. It has a zero object.

  3. It has kernels and cokernels, images and coimages.

  4. In general it is not an abelian category.

Proof. It is clear that $\text{Fil}(\mathcal{A})$ is additive with direct sum given by $(A, F) \oplus (B, F) = (A \oplus B, F)$ where $F^ p(A \oplus B) = F^ pA \oplus F^ pB$. The kernel of a morphism $f : (A, F) \to (B, F)$ of filtered objects is the injection $\mathop{\mathrm{Ker}}(f) \subset A$ where $\mathop{\mathrm{Ker}}(f)$ is endowed with the induced filtration. The cokernel of a morphism $f : A \to B$ of filtered objects is the surjection $B \to \mathop{\mathrm{Coker}}(f)$ where $\mathop{\mathrm{Coker}}(f)$ is endowed with the quotient filtration. Since all kernels and cokernels exist, so do all coimages and images. See Example 12.3.13 for the last statement. $\square$

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