The singular antidominant Serre quotient

Since ${\mathrm{\Gamma }}_{\kappa }$ is exact, its kernel $K$ is closed under subobjects, quotients and extensions; hence $K$ is a Serre subcategory. The functor ${\mathrm{\Gamma }}_{\kappa }$ therefore descends to the Serre quotient.

Localization is right exact. For a short exact sequence $0\to V^{\mathrm{\prime }}\to V\to V^{\mathrm{\prime }\mathrm{\prime }}\to 0$ of $A_{\kappa }$-modules, applying localization gives a right-exact sequence $\mathrm{Loc}{V}^{\mathrm{\prime }}⟶\mathrm{Loc}V⟶\mathrm{Loc}{V}^{\mathrm{\prime }\mathrm{\prime }}⟶0.$ Apply the exact functor ${\mathrm{\Gamma }}_{\kappa }$ and use [F2]. The resulting sequence is $0\to V^{\mathrm{\prime }}\to V\to V^{\mathrm{\prime }\mathrm{\prime }}\to 0$. Therefore the kernel of $\mathrm{Loc}{V}^{\mathrm{\prime }}\to \mathrm{Loc}V$ has zero global sections and lies in $K$. Thus localization becomes exact after passage to the Serre quotient.

The unit is an isomorphism by [F2], so it remains an isomorphism in the quotient. For any $M$, the triangular identity in [F3] shows that applying ${\mathrm{\Gamma }}_{\kappa }$ to the counit ${\mathrm{Loc}}_{\kappa }{\mathrm{\Gamma }}_{\kappa }M\to M$ gives an isomorphism. By exactness [F1], the kernel and cokernel of the counit lie in $K$; hence the counit is an isomorphism in the quotient. The descended unit and counit prove the claimed equivalence.

For an arbitrary $M$, let $E$ be the image of the counit ${\mathrm{Loc}}_{\kappa }\mathrm{\Gamma }M\to M$. By step 3.1 its cokernel has zero global sections, so $E$ and $M$ define isomorphic objects of the Serre quotient. The source is generated by its global sections, hence so is $E$. This holds even for generation as an ${\mathcal{O}}_X$-module: PBW and the surjective action anchor show that ${\mathcal{D}}_{\kappa }$ is locally spanned over ${\mathcal{O}}_X$ by products of the global $\mathfrak{g}$-operators; their actions on global sections remain global sections. Among the submodules of $E$ with zero global sections, take their sum $T$. This is a set-indexed sum: a submodule sheaf is specified by a family of subsets of the sets $E(U)$, one for each open $U\subseteq X$. The direct sum of any such family is again a quasicoherent ${\mathcal{D}}_{\kappa }$-module, and $\mathrm{\Gamma }$ commutes with it by [F4]. Exactness of $\mathrm{\Gamma }$ [F1] then gives $\mathrm{\Gamma }T=0$, so $T$ is the largest submodule of $E$ in $K$. The quotient $E\mathrm{/}T$ remains generated by global sections. If $S\subseteq E\mathrm{/}T$ has $\mathrm{\Gamma }S=0$, its inverse image $P\subseteq E$ fits into $0\to T\to P\to S\to 0$; exactness [F1] gives $\mathrm{\Gamma }P=0$, so $P\subseteq T$ by maximality and $S=0$. Thus $E\mathrm{/}T$ is a representative of the same quotient object, is generated by global sections, and has no nonzero submodule in $K$. Conversely, every module with those two properties is by definition such a representative.

These representatives give an actual equivalence with the algebra-module category, not just a choice of quotient objects. If $E,D$ are generated by global sections and have no nonzero subobject in $K$, any map with zero global sections has image in $K$ and hence is zero. Given $h:\mathrm{\Gamma }E\to \mathrm{\Gamma }D$, adjunction [F3] gives $f:\mathrm{Loc}\mathrm{\Gamma }E\to D$. The evaluation counit onto $E$ is surjective because $E$ is generated by global sections; its kernel is in $K$ by step 3.1. Its image under $f$ is therefore a $K$-subobject of $D$ and is zero, so $f$ factors uniquely through a map $E\to D$ inducing $h$. This proves full faithfulness of $\mathrm{\Gamma }$ on these representatives. For any algebra module $V$, apply the construction of step 4.1 to $\mathrm{Loc}V$; [F1]–[F2] give its representative global sections equal to $V$, proving essential surjectivity.