Let $\mathcal A$ be an abelian category with enough injectives. Let $\mathcal I$ be the collection of injective objects. Let $\mathcal A/\mathcal I$ be the quotient category whose objects are same as those of $\mathcal A$ and morphisms are $\operatorname{Hom}_{\mathcal A/\mathcal I}(X,Y)/R_{X,Y}$, where $R_{X,Y}$, where $R_{X,Y}$ is the collection of all morphisms factoring through some object of $\mathcal I$. It is well-known that $\mathcal A/\mathcal I$ has a natural right triangulated structure, see Theorem 2.12(ii) of Beligiannis–Marmaridis 1994 (DOI link). Consider the stabilization $S(\mathcal A/\mathcal I)$ as defined in Remark 3.2 of Beligiannis 2000 (DOI link).

My question is: I wonder if there is an analogue of Theorem 3.8 of Belligianis 2000 (previous link) in the above setting? In particular, I wonder if $S(\mathcal A/\mathcal I) \cong D^b(\mathcal A)/K^b(\mathcal I)$ ?

Let $\mathcal A$ be an abelian category with enough injectives. Let $\mathcal I$ be the collection of injective objects. Let $\mathcal A/\mathcal I$ be the quotient category whose objects are same as those of $\mathcal A$ and morphisms are $\operatorname{Hom}_{\mathcal A/\mathcal I}(X,Y)/R_{X,Y}$, where $R_{X,Y}$, where $R_{X,Y}$ is the collection of all morphisms factoring through some object of $\mathcal I$. It is well-known that $\mathcal A/\mathcal I$ has a natural right triangulated structure, see Theorem 2.12(ii) of Beligiannis–Marmaridis 1994 (DOI link). Consider the stabilization $S(\mathcal A/\mathcal I)$ as defined in Remark 3.2 of Beligiannis 2000 (DOI link).

My question is: I wonder if there is an analogue of Theorem 3.8 of Beligiannis 2000 (previous link) in the above setting? In particular, I wonder if $S(\mathcal A/\mathcal I) \cong D^b(\mathcal A)/K^b(\mathcal I)$ ?

Let $\mathcal A$ be an abelian category with enough injectives. Let $\mathcal I$ be the collection of injective objects. Let $\mathcal A/\mathcal I$ be the quotient category whose objects are same as those of $\mathcal A$ and morphisms are $Hom_{\mathcal A/\mathcal I}(X,Y)/R_{X,Y}$, where $R_{X,Y}$, where $R_{X,Y}$ is the collection of all morphisms factoring through some object of $\mathcal I$. It is well-known that $\mathcal A/\mathcal I$ has a natural right triangulated structure, see Theorem 2.12(ii) of https://doi.org/10.1080/00927879408825119. Consider the stabilization $S(\mathcal A/\mathcal I)$ as defined in Remark 3.2 of https://doi.org/10.1080/00927870008827105.

My question is: I wonder if there is an analogue of Theorem 3.8 of https://doi.org/10.1080/00927870008827105 in the above setting? In particular, I wonder if $S(\mathcal A/\mathcal I) \cong D^b(\mathcal A)/K^b(\mathcal I)$ ?

Let $\mathcal A$ be an abelian category with enough injectives. Let $\mathcal I$ be the collection of injective objects. Let $\mathcal A/\mathcal I$ be the quotient category whose objects are same as those of $\mathcal A$ and morphisms are $\operatorname{Hom}_{\mathcal A/\mathcal I}(X,Y)/R_{X,Y}$, where $R_{X,Y}$, where $R_{X,Y}$ is the collection of all morphisms factoring through some object of $\mathcal I$. It is well-known that $\mathcal A/\mathcal I$ has a natural right triangulated structure, see Theorem 2.12(ii) of Beligiannis–Marmaridis 1994 (DOI link). Consider the stabilization $S(\mathcal A/\mathcal I)$ as defined in Remark 3.2 of Beligiannis 2000 (DOI link).

My question is: I wonder if there is an analogue of Theorem 3.8 of Belligianis 2000 (previous link) in the above setting? In particular, I wonder if $S(\mathcal A/\mathcal I) \cong D^b(\mathcal A)/K^b(\mathcal I)$ ?