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The unit V to Gamma Loc_kappa(V) is an isomorphism for antidominant twists

Statement

For antidominant $κ$ and every left $A_{κ}$ -module $V$ , the unit of the adjunction in Localization is left adjoint to global sections is an isomorphism $V \tilde{\to} Γ (X, {Loc}_{κ} V) .$

Facts & Assumptions

Given: Antidominant $κ$ and a left $A_{κ}$ -module $V$ .

[F1]

$Γ (X, -)$ is exact on quasi-coherent $D_{κ}$ -modules for antidominant $κ$ (Antidominant twists are D-affine and regular ones are conservative).

[F2]

$Γ (X, D_{κ}) = A_{κ}$ as a filtered algebra (Global sections of twisted differential operators on the flag variety).

[F3]

On this finite affine cover, global sections of quasi-coherent modules commute with filtered colimits, hence with arbitrary direct sums (Quasicoherent modules are filtered unions of coherent subsheaves and cohomology commutes with these colimits).

Proof

1.1F1given

Choose a free presentation $A_{κ}^{(I)} ⟶ A_{κ}^{(J)} ⟶ V ⟶ 0.$ Localization is a tensor functor, so it is right exact and gives $D_{κ}^{(I)} ⟶ D_{κ}^{(J)} ⟶ {Loc}_{κ} (V) ⟶ 0.$ Applying the exact global-sections functor [F1] preserves exactness.

2.1F2F3step 1.1algebra∎

By [F3], global sections commute with the direct sums in step 1.1; by [F2], each free term becomes the corresponding free $A_{κ}$ -module. Under the adjunction unit, the maps on these free terms are the identity maps on $A_{κ}^{(I)}$ and $A_{κ}^{(J)}$ . The cokernels of the two presentations are therefore identified by the induced unit map on $V$ . Hence $V \to Γ (X, {Loc}_{κ} V)$ is an isomorphism.

Depends on

Used by

Dependency tree · two levels

36 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Masaki Kashiwara, Representation theory and D-modules on flag varieties, Astérisque 173–174 (1989) (standard reference, not scraped)
Yi Sun, D-modules and representation theory (standard reference, not scraped)