Regular antidominant Beilinson–Bernstein localization

The unit is an isomorphism by [F2]. For a quasi-coherent ${\mathcal{D}}_{\kappa }$-module $M$, let ${ϵ}_M:{\mathrm{Loc}}_{\kappa }{\mathrm{\Gamma }}_{\kappa }M\to M$ be the counit. The triangular identity for the adjunction says ${\mathrm{\Gamma }}_{\kappa }({ϵ}_M)\circ {\eta }_{{\mathrm{\Gamma }}_{\kappa }M}={\mathrm{id}}_{{\mathrm{\Gamma }}_{\kappa }M},$ where $\eta$ is the unit. Since $\eta$ is an isomorphism, $\mathrm{\Gamma }({ϵ}_M)$ is an isomorphism.

Exactness of $\mathrm{\Gamma }$ in [F3] implies $\mathrm{\Gamma }(\ker {ϵ}_M)=0$ and $\mathrm{\Gamma }(\mathrm{coker}{ϵ}_M)=0$. Conservativity in [F3] makes both sheaves zero, so ${ϵ}_M$ is an isomorphism. The unit and counit are isomorphisms, hence the adjoint functors are inverse equivalences.

By Noetherianity of the enveloping algebra, $U(\mathfrak{g})$ is left Noetherian, so its quotient $A_{\kappa }$ is left Noetherian. On a smooth affine chart of $X$, $\mathrm{Sym}{T}_X$ is a polynomial algebra over a finite-type $\mathbb{C}$-algebra and hence is Noetherian by If $R$ is Noetherian then $R[x_1,\dots ,x_n]$ is Noetherian for every $n\in \mathbb{N}$. If a filtered ring has Noetherian associated graded, every left ideal is finitely generated: choose finitely many lifts of homogeneous generators of its graded ideal, subtract their leading terms from an arbitrary ideal element, and induct on its finite filtration degree. This proves that ${\mathcal{D}}_{\kappa }$ is locally left Noetherian. A finitely generated $A_{\kappa }$-module has a finite presentation over the Noetherian ring $A_{\kappa }$; localization presents it locally as a quotient of a finite free ${\mathcal{D}}_{\kappa }$-module, hence as a coherent ${\mathcal{D}}_{\kappa }$-module.

Conversely, let $M$ be coherent over ${\mathcal{D}}_{\kappa }$. A finite affine cover of $X$ and local finite generation give finitely many local generators. The counit is an isomorphism by step 2.1, so each such generator is locally in the image of the evaluation map ${\mathcal{D}}_{\kappa }{\otimes }_{\mathbb{C}}\mathrm{\Gamma }(X,M)\to M$. These local image neighborhoods cover $X$; quasi-compactness gives a finite subcover, hence a finite set $S\subset \mathrm{\Gamma }(X,M)$ whose ${\mathcal{D}}_{\kappa }$-span is all of $M$. Thus the map ${\mathcal{D}}_{\kappa }^{\mathrm{\mid }S\mathrm{\mid }}\to M$ is surjective. Exactness of $\mathrm{\Gamma }$ in [F3] gives a surjection $A_{\kappa }^{\mathrm{\mid }S\mathrm{\mid }}=\mathrm{\Gamma }(X,{\mathcal{D}}_{\kappa }^{\mathrm{\mid }S\mathrm{\mid }})\to \mathrm{\Gamma }(X,M)$, so $\mathrm{\Gamma }(X,M)$ is finitely generated over $A_{\kappa }$. This is coherence over ${\mathcal{D}}_{\kappa }$, not coherence over ${\mathcal{O}}_X$.