# Do we have criteria of strict localization of a Grothendieck category?

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?