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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}.

Using transfinite recursion 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 (3)

Comment #848 by on

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

Comment #9522 by on

As in other proofs, we are using AB5 to get that is injective for a limit ordinal and , by means of the lemma in Comment #9497.

To assure the existence of with one could invoke Sets, Proposition 3.7.2.

Comment #9816 by on

An alternative reference for this result is Kashiwara, Schapira, Categories and Sheaves, Corollary 14.1.8.


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