Lemma 13.26.12. Let \mathcal{A} be an abelian category with enough injectives. Let \mathcal{I}^ f \subset \text{Fil}^ f(\mathcal{A}) denote the strictly full additive subcategory whose objects are the filtered injective objects. The canonical functor
K^{+}(\mathcal{I}^ f) \longrightarrow DF^{+}(\mathcal{A})
is exact, fully faithful and essentially surjective, i.e., an equivalence of triangulated categories. Furthermore the diagrams
\xymatrix{ K^{+}(\mathcal{I}^ f) \ar[d]_{\text{gr}^ p} \ar[r] & DF^{+}(\mathcal{A}) \ar[d]_{\text{gr}^ p} \\ K^{+}(\mathcal{I}) \ar[r] & D^{+}(\mathcal{A}) } \quad \xymatrix{ K^{+}(\mathcal{I}^ f) \ar[d]^{\text{forget }F} \ar[r] & DF^{+}(\mathcal{A}) \ar[d]^{\text{forget }F} \\ K^{+}(\mathcal{I}) \ar[r] & D^{+}(\mathcal{A}) }
are commutative, where \mathcal{I} \subset \mathcal{A} is the strictly full additive subcategory whose objects are the injective objects.
Comments (0)