The Stacks project

The limit of a “split” tower of K-injective complexes is K-injective.

Lemma 13.31.8. Let $\mathcal{A}$ be an abelian category. Let

\[ \ldots \to I_3^\bullet \to I_2^\bullet \to I_1^\bullet \]

be an inverse system of complexes. Assume

  1. each $I_ n^\bullet $ is $K$-injective,

  2. each map $I_{n + 1}^ m \to I_ n^ m$ is a split surjection,

  3. the limits $I^ m = \mathop{\mathrm{lim}}\nolimits I_ n^ m$ exist.

Then the complex $I^\bullet $ is K-injective.

Proof. We urge the reader to skip the proof of this lemma. Let $M^\bullet $ be an acyclic complex. Let us abbreviate $H_ n(a, b) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(M^ a, I_ n^ b)$. With this notation $\mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A})}(M^\bullet , I^\bullet )$ is the cohomology of the complex

\[ \prod _ m \mathop{\mathrm{lim}}\nolimits \limits _ n H_ n(m, m - 2) \to \prod _ m \mathop{\mathrm{lim}}\nolimits \limits _ n H_ n(m, m - 1) \to \prod _ m \mathop{\mathrm{lim}}\nolimits \limits _ n H_ n(m, m) \to \prod _ m \mathop{\mathrm{lim}}\nolimits \limits _ n H_ n(m, m + 1) \]

in the third spot from the left. We may exchange the order of $\prod $ and $\mathop{\mathrm{lim}}\nolimits $ and each of the complexes

\[ \prod _ m H_ n(m, m - 2) \to \prod _ m H_ n(m, m - 1) \to \prod _ m H_ n(m, m) \to \prod _ m H_ n(m, m + 1) \]

is exact by assumption (1). By assumption (2) the maps in the systems

\[ \ldots \to \prod _ m H_3(m, m - 2) \to \prod _ m H_2(m, m - 2) \to \prod _ m H_1(m, m - 2) \]

are surjective. Thus the lemma follows from Homology, Lemma 12.31.4. $\square$

Comments (3)

Comment #853 by Bhargav Bhatt on

Suggested slogan: The limit of a "split" tower of K-injective complexes is K-injective.

Comment #4337 by Manuel Hoff on

In the formulation of the Lemma, it is mentioned two times that the complexes are K-injective (one time in the second sentence and a second time in (1)).

There are also:

  • 5 comment(s) on Section 13.31: K-injective complexes

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Your email address will not be published.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 070L. Beware of the difference between the letter 'O' and the digit '0'.