Lemma 13.23.3. Let \mathcal{A} be an abelian category with enough injectives. Given a resolution functor (j, i) there is a unique way to turn j into a functor and i into a 2-isomorphism producing a 2-commutative diagram
\xymatrix{ K^{+}(\mathcal{A}) \ar[rd] \ar[rr]_ j & & K^{+}(\mathcal{I}) \ar[ld] \\ & D^{+}(\mathcal{A}) }
where \mathcal{I} is the full additive subcategory of \mathcal{A} consisting of injective objects.
Proof.
For every morphism \alpha : K^\bullet \to L^\bullet of K^{+}(\mathcal{A}) there is a unique morphism j(\alpha ) : j(K^\bullet ) \to j(L^\bullet ) in K^{+}(\mathcal{I}) such that
\xymatrix{ K^\bullet \ar[r]_\alpha \ar[d]_{i_{K^\bullet }} & L^\bullet \ar[d]^{i_{L^\bullet }} \\ j(K^\bullet ) \ar[r]^{j(\alpha )} & j(L^\bullet ) }
is commutative in K^{+}(\mathcal{A}). To see this either use Lemmas 13.18.6 and 13.18.7 or the equivalent Lemma 13.18.8. The uniqueness implies that j is a functor, and the commutativity of the diagram implies that i gives a 2-morphism which witnesses the 2-commutativity of the diagram of categories in the statement of the lemma.
\square
Comments (3)
Comment #8420 by Elías Guisado on
Comment #9044 by Stacks project on
Comment #9473 by Elías Guisado on
There are also: