Lemma 12.19.10 (05SM)—The Stacks project

Lemma 12.19.10. Let $\mathcal{A}$ be an abelian category. Let $A, B, C \in \text{Fil}(\mathcal{A})$ . Let $f : A \to B$ and $g : A \to C$ be morphisms. Then there exists a pushout

$\xymatrix{ A \ar[r]_ f \ar[d]_ g & B \ar[d]^{g'} \\ C \ar[r]^{f'} & C \amalg _ A B }$

in $\text{Fil}(\mathcal{A})$ . If $f$ is strict, so is $f'$ .

Proof. Set $C \amalg _ A B$ equal to $\mathop{\mathrm{Coker}}((g, -f) : A \to C \oplus B)$ in $\text{Fil}(\mathcal{A})$ . This cokernel exists, by Lemma 12.19.2. It is a pushout, see Example 12.5.6. Note that $F^ p(C \amalg _ A B)$ is the image of $F^ pC \oplus F^ pB$ . Hence

$(f')^{-1}(F^ p(C \amalg _ A B)) = g(f^{-1}(F^ pB))) + F^ pC$

Whence the last statement. $\square$

Comments (5)

Comment #3762 by Owen B on October 31, 2018 at 20:43

just a typo: $\textstyle\prod_A$ appears twice in place of $\coprod_A$

Comment #3763 by Owen B on October 31, 2018 at 20:44

sorry, I meant $\times_A$ appears twice in place of $\coprod_A$

Comment #3894 by Johan on January 22, 2019 at 10:05

Thanks and fixed here.

Comment #8754 by Colin Ni on December 25, 2023 at 08:37

In the first sentence, I believe the $(1, -1)$ should be $(f, -g)$

Comment #9323 by Stacks project on May 31, 2024 at 14:40

OK, I think it should be $(g, -f)$ . Fixed here.

There are also:

6 comment(s) on Section 12.19: Filtrations

Comments (5)

Post a comment