Gabriel–Popescu theorem

Remarks

The functor $T$ is the colimit-preserving functor with $T(R) = U$, often written $T(M) = M \otimes_R U$. The hard part of the proof is the exactness of $T$; it uses that filtered colimits are exact in a Grothendieck category and that $U$ generates. As a consequence, $\mathcal{C}$ is equivalent to the quotient of $\mathrm{Mod}\text{-}R$ by the Serre subcategory of modules $M$ with $T(M) = 0$. That subcategory is closed under arbitrary direct sums, so it is a localizing subcategory. Example: for $\mathcal{C} = \mathrm{Mod}\text{-}S$ and $U = S$, one gets $R \cong S$, the functor $G$ is an equivalence, and the localizing subcategory is $0$. Example: the category of abelian sheaves on a topological space is a Grothendieck category, so the theorem presents it as a quotient of a module category.

Going deeper

For a student

The theorem is an analogue of the Freyd–Mitchell embedding theorem for large categories with a generator. The embedding $G$ is left exact but in general not exact; in exchange, it has an exact left adjoint, so $\mathcal{C}$ is a reflective subcategory of $\mathrm{Mod}\text{-}R$ with exact reflector. This makes it a basic tool for transferring module-theoretic results to Grothendieck categories. It is also the starting point of the description of Grothendieck categories as localizations of module categories.