The Stacks project

Lemma 13.26.5. Let $\mathcal{A}$ be an abelian category with enough injectives. For any object $A$ of $\text{Fil}^ f(\mathcal{A})$ there exists a strict monomorphism $A \to I$ where $I$ is a filtered injective object.

Proof. Pick $a \leq b$ such that $\text{gr}^ p(A) = 0$ unless $p \in \{ a, a + 1, \ldots , b\} $. For each $n \in \{ a, a + 1, \ldots , b\} $ choose an injection $u_ n : A/F^{n + 1}A \to I_ n$ with $I_ n$ an injective object. Set $I = \bigoplus _{a \leq n \leq b} I_ n$ with filtration $F^ pI = \bigoplus _{n \geq p} I_ n$ and set $u : A \to I$ equal to the direct sum of the maps $u_ n$. $\square$

Comments (2)

Comment #3766 by Owen B on

typos: an injective object, not and , not additionally, it appears that the indexing is off. I think it is fixed if you let rather than what is written.

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05TS. Beware of the difference between the letter 'O' and the digit '0'.