The Stacks project

Existence of K-injective complexes for Grothendieck abelian categories.

Theorem 19.12.6. Let $\mathcal{A}$ be a Grothendieck abelian category. For every complex $M^\bullet $ there exists a quasi-isomorphism $M^\bullet \to I^\bullet $ such that $M^ n \to I^ n$ is injective and $I^ n$ is an injective object of $\mathcal{A}$ for all $n$ and $I^\bullet $ is a K-injective complex. Moreover, the construction is functorial in $M^\bullet $.

Proof. Please compare with the proof of Theorem 19.2.8 and Theorem 19.11.7. Choose a cardinal $\kappa $ as in Lemmas 19.12.2 and 19.12.3. Choose a set $(K_ i^\bullet )_{i \in I}$ of bounded above, acyclic complexes such that every bounded above acyclic complex $K^\bullet $ such that $|K^ n| \leq \kappa $ is isomorphic to $K_ i^\bullet $ for some $i \in I$. This is possible by Lemma 19.11.4. Denote $\mathbf{M}^\bullet (-)$ the functor constructed in Lemma 19.12.4. Denote $\mathbf{N}^\bullet (-)$ the functor constructed in Lemma 19.12.5. Both of these functors come with injective transformations $\text{id} \to \mathbf{M}$ and $\text{id} \to \mathbf{N}$.

By transfinite induction we define a sequence of functors $\mathbf{T}_\alpha (-)$ and corresponding transformations $\text{id} \to \mathbf{T}_\alpha $. Namely we set $\mathbf{T}_0(M^\bullet ) = M^\bullet $. If $\mathbf{T}_\alpha $ is given then we set

\[ \mathbf{T}_{\alpha + 1}(M^\bullet ) = \mathbf{N}^\bullet (\mathbf{M}^\bullet (\mathbf{T}_\alpha (M^\bullet ))) \]

If $\beta $ is a limit ordinal we set

\[ \mathbf{T}_\beta (M^\bullet ) = \mathop{\mathrm{colim}}\nolimits _{\alpha < \beta } \mathbf{T}_\alpha (M^\bullet ) \]

The transition maps of the system are injective quasi-isomorphisms. By AB5 we see that the colimit is still quasi-isomorphic to $M^\bullet $. We claim that $M^\bullet \to \mathbf{T}_\alpha (M^\bullet )$ does the job if the cofinality of $\alpha $ is larger than $\max (\kappa , |U|)$ where $U$ is a generator of $\mathcal{A}$. Namely, it suffices to check conditions (1) and (2) of Lemma 19.12.3.

For (1) we use the criterion of Lemma 19.11.6. Suppose that $M \subset U$ and $\varphi : M \to \mathbf{T}^ n_\alpha (M^\bullet )$ is a morphism for some $n \in \mathbf{Z}$. By Proposition 19.11.5 we see that $\varphi $ factor through $\mathbf{T}^ n_{\alpha '}(M^\bullet )$ for some $\alpha ' < \alpha $. In particular, by the construction of the functor $\mathbf{N}^\bullet (-)$ we see that $\varphi $ factors through an injective object of $\mathcal{A}$ which shows that $\varphi $ lifts to a morphism on $U$.

For (2) let $w : K^\bullet \to \mathbf{T}_\alpha (M^\bullet )$ be a morphism of complexes where $K^\bullet $ is a bounded above acyclic complex such that $|K^ n| \leq \kappa $. Then $K^\bullet \cong K_ i^\bullet $ for some $i \in I$. Moreover, by Proposition 19.11.5 once again we see that $w$ factor through $\mathbf{T}^ n_{\alpha '}(M^\bullet )$ for some $\alpha ' < \alpha $. In particular, by the construction of the functor $\mathbf{M}^\bullet (-)$ we see that $w$ is homotopic to zero. This finishes the proof. $\square$


Comments (1)

Comment #848 by on

Suggested slogan: Existence of K-injective complexes for Grothendieck abelian categories.


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 079P. Beware of the difference between the letter 'O' and the digit '0'.