Let $\mathcal{C}$ be an abelian category and $\mathcal{S}$ be a full subcategory of $\mathcal{C}$. We call $\mathcal{S}$ a Serre subcategory of $\mathcal{C}$ if $\mathcal{S}$ is closed under forming subobjects, quotients, and extensions. For a Serre subcategory $\mathcal{S}$ we could form the quotien category $\mathcal{C}/\mathcal{S}$ and have the quotient functor $p: \mathcal{C}\to \mathcal{C}/\mathcal{S}$, which is obviously exact.

We call $\mathcal{S}$ a localizing Serre subcategory of $\mathcal{C}$ if the quotient functor $p: \mathcal{C}\to \mathcal{C}/\mathcal{S}$ has a right adjoint: $s: \mathcal{C}/\mathcal{S}\to \mathcal{C}$. In this case we could prove that $ps$ is isomorphic to the identity functor. In particular $s: \mathcal{C}/\mathcal{S}\to \mathcal{C}$ is an embedding hence $\mathcal{C}/\mathcal{S}$ is equivalent to a full subcategory $\mathcal{L}$ of $\mathcal{C}$. Actually $\mathcal{L}$ is the right orthogonal complement of $\mathcal{S}$.

Similarly we call a full subcategory $\mathcal{L}$ of an abelian category $\mathcal{C}$ a strict localization if the inclusion functor $i: \mathcal{L}\to \mathcal{C}$ has an exact left adjoint $a: \mathcal{C}\to \mathcal{L}$. Notice that in general the inclusion $i: \mathcal{L}\to \mathcal{C}$ is not an exact functor.

In the case that $\mathcal{C}$ is a Grothendieck category, we know that a Serre subcategory $\mathcal{S}$ of $\mathcal{C}$ is localizing if and only if $\mathcal{S}$ is closed under arbitrary direct sums. See this paper, Proposition 2.5.

My question is: When $\mathcal{C}$ is a Grothendieck category, do we have an explicit criterion on what kind of full subcategory $\mathcal{L}$ of $\mathcal{C}$ is a strict localization?


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